TPTP Problem File: NUM924

TPTP Problem File: NUM924_4.p

%------------------------------------------------------------------------------
% File     : NUM924_4 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Number Theory
% Problem  : Sum of two squares line 102, 5000 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : s2s_5000_tff_l102 [Bla11]

% Status   : Theorem
% Rating   : 0.11 v8.2.0, 0.20 v8.1.0, 0.45 v7.5.0, 0.50 v7.4.0, 0.20 v7.3.0, 0.17 v7.1.0, 0.00 v6.2.0, 0.67 v6.1.0, 0.33 v6.0.0, 0.40 v5.5.0, 0.25 v5.4.0, 0.33 v5.3.0
% Syntax   : Number of formulae    : 7017 (2347 unt;1596 typ;   0 def)
%            Number of atoms       : 11701 (4354 equ)
%            Maximal formula atoms :   17 (   1 avg)
%            Number of connectives : 7207 ( 927   ~; 197   |; 697   &)
%                                         ( 908 <=>;4478  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   4 avg)
%            Maximal term depth    :   21 (   2 avg)
%            Number of types       :  525 ( 524 usr)
%            Number of type conns  : 1195 ( 647   >; 548   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    : 1071 (1071 usr; 425 con; 0-8 aty)
%            Number of variables   : 11604 (11333   !; 271   ?;11604   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 19:06:15
%------------------------------------------------------------------------------
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__p_202,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc___204,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J,type,
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tff(ty_ty_tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Rat__Orat_J,type,
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tff(ty_ty_tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
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tff(ty_ty_tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickcheck____Nar,type,
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tff(ty_ty_tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
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%----Explicit typings (1072)
tff(sy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Int__Oint_000tc__In,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__In,type,
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tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__Na,type,
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tff(sy_c_BijectionRel_ObijR_000tc__Int__Oint_000tc__Int__Oint,type,
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tff(sy_c_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__221,type,
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tff(sy_c_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat,type,
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tff(sy_c_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool,type,
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    cOMBS_nat_bool_bool: fun_fu1901564496t_bool ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat,type,
    cOMBS_nat_nat_nat: fun_fu1382375754at_nat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat,type,
    cOMBS_nat_rat_rat: fun_fu941266250at_rat ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocomplex,type,
    cOMBS_696648398omplex: fun_fu1913390252omplex ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    cOMBS_nat_real_real: fun_fu247819600t_real ).

tff(sy_c_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal,type,
    cOMBS_337378985l_real: fun_fu385122018l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_real_bool_bool: fun_fu1882383245l_bool ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    cOMBS_real_real_real: fun_fu647765353l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef_,type,
    cOMBS_511527878l_real: fun_fu1698495721l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Or,type,
    cOMBS_193881978l_real: fun_fu1613343649l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__R,type,
    cOMBS_107495110l_real: fun_fu1574854795l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__R_245,type,
    cOMBS_163651580l_real: fun_fu1943305629l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__O,type,
    cOMBS_794971624l_real: fun_fu209750539l_real ).

tff(sy_c_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__O_246,type,
    cOMBS_902912708l_real: fun_fu2079587053l_real ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool_000tc__,type,
    cOMBS_400904860l_bool: fun_fu1257888773t_bool ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__H,type,
    cOMBS_480885903t_bool: fun_fu581830891t_bool ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__I,type,
    cOMBS_2019229620nt_int: fun_fu1660428975nt_int ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint_000tc__R,type,
    cOMBS_549334880nt_rat: fun_fu1887196079nt_rat ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc_,type,
    cOMBS_1610675060t_bool: fun_fu620454517t_bool ).

tff(sy_c_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc__247,type,
    cOMBS_1676841879t_bool: fun_fu2062813463t_bool ).

tff(sy_c_Code__Numeral_OSuc__code__numeral,type,
    code_S1047413653umeral: code_code_numeral > code_code_numeral ).

tff(sy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__size,type,
    code_c271388182l_size: code_code_numeral > nat ).

tff(sy_c_Code__Numeral_Odiv__mod__code__numeral,type,
    code_d418564891umeral: ( code_code_numeral * code_code_numeral ) > produc1359518119umeral ).

tff(sy_c_Code__Numeral_Oint__of,type,
    code_int_of: code_code_numeral > int ).

tff(sy_c_Code__Numeral_Onat__of__aux,type,
    code_nat_of_aux: code_code_numeral > fun_nat_nat ).

tff(sy_c_Complete__Lattice_OSup__class_OSup_000tc__RealDef__Oreal,type,
    comple124823625p_real: fun_real_bool > real ).

tff(sy_c_Complete__Lattice_OSup__class_OSup_000tc__fun_Itc__Int__Oint_Mtc__HOL__Oboo,type,
    comple219730294t_bool: fun_fu819804446l_bool > fun_int_bool ).

tff(sy_c_Complex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Oarg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: fun_complex_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex_1: fun_re398676606omplex ).

tff(sy_c_Complex_Ocomplex_Ocomplex__size,type,
    complex_size: complex > nat ).

tff(sy_c_Complex_Oexpi,type,
    expi: complex > complex ).

tff(sy_c_Complex_Oii,type,
    ii: complex ).

tff(sy_c_Complex_Orcis,type,
    rcis: ( real * real ) > complex ).

tff(sy_c_Deriv_OBolzano__bisect,type,
    bolzano_bisect: ( fun_Pr1127657839l_bool * real * real ) > fun_na1745536976l_real ).

tff(sy_c_Deriv_Oderiv_000tc__RealDef__Oreal,type,
    deriv_real: ( fun_real_real * real ) > fun_real_bool ).

tff(sy_c_Divides_Oadjust,type,
    adjust: int > fun_Pr241715507nt_int ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Code____Numeral__Ocode____numeral,type,
    div_di1218280263umeral: fun_Co1323339824umeral ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Int__Oint,type,
    div_div_int: fun_int_fun_int_int ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Nat__Onat,type,
    div_div_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Divides_Odiv__class_Odiv_000tc__Quickcheck____Narrowing__Ocode____int,type,
    div_di1430059507de_int: fun_Qu64793220de_int ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Code____Numeral__Ocode____numeral,type,
    div_mo1740067990umeral: code_code_numeral > fun_Co536835517umeral ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Int__Oint,type,
    div_mod_int: int > fun_int_int ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Nat__Onat,type,
    div_mod_nat: nat > fun_nat_nat ).

tff(sy_c_Divides_Odiv__class_Omod_000tc__Quickcheck____Narrowing__Ocode____int,type,
    div_mo231679042de_int: quickcheck_code_int > fun_Qu812286141de_int ).

tff(sy_c_Divides_Odivmod__int,type,
    divmod_int: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_Odivmod__int__rel,type,
    divmod_int_rel: ( int * int ) > fun_Pr974702441t_bool ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Divides_Odivmod__nat__rel,type,
    divmod_nat_rel: ( nat * nat ) > fun_Pr1927655017t_bool ).

tff(sy_c_Divides_OnegDivAlg,type,
    negDivAlg: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_OnegDivAlg__rel,type,
    negDivAlg_rel: fun_Pr313471731t_bool ).

tff(sy_c_Divides_OnegateSnd,type,
    negateSnd: fun_Pr241715507nt_int ).

tff(sy_c_Divides_Opdivmod,type,
    pdivmod: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_OposDivAlg,type,
    posDivAlg: ( int * int ) > product_prod_int_int ).

tff(sy_c_Divides_OposDivAlg__rel,type,
    posDivAlg_rel: fun_Pr313471731t_bool ).

tff(sy_c_EulerFermat_OBnorRset,type,
    bnorRset: int > fun_int_fun_int_bool ).

tff(sy_c_EulerFermat_ORRset2norRR,type,
    rRset2norRR: ( fun_int_bool * int ) > fun_int_int ).

tff(sy_c_EulerFermat_ORsetR,type,
    rsetR: int > fun_fu819804446l_bool ).

tff(sy_c_EulerFermat_Ois__RRset,type,
    is_RRset: fun_int_bool > fun_int_bool ).

tff(sy_c_EulerFermat_OnoXRRset,type,
    noXRRset: int > fun_int_fun_int_bool ).

tff(sy_c_EulerFermat_OnorRRset,type,
    norRRset: fun_int_fun_int_bool ).

tff(sy_c_EulerFermat_Ophi,type,
    phi: fun_int_nat ).

tff(sy_c_EulerFermat_Ozcongm,type,
    zcongm: int > fun_int_fun_int_bool ).

tff(sy_c_Euler_OMultInvPair,type,
    multInvPair: ( int * int ) > fun_int_fun_int_bool ).

tff(sy_c_Euler_OSetS,type,
    setS: ( int * int ) > fun_fu819804446l_bool ).

tff(sy_c_EvenOdd_OzEven,type,
    zEven: fun_int_bool ).

tff(sy_c_EvenOdd_OzOdd,type,
    zOdd: fun_int_bool ).

tff(sy_c_Fact_Ofact__class_Ofact_000tc__Int__Oint,type,
    fact_fact_int: fun_int_int ).

tff(sy_c_Fact_Ofact__class_Ofact_000tc__Nat__Onat,type,
    fact_fact_nat: fun_nat_nat ).

tff(sy_c_Fields_Oinverse__class_Odivide_000tc__Complex__Ocomplex,type,
    invers1025623611omplex: fun_co822197010omplex ).

tff(sy_c_Fields_Oinverse__class_Odivide_000tc__Rat__Orat,type,
    inverse_divide_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Fields_Oinverse__class_Odivide_000tc__RealDef__Oreal,type,
    inverse_divide_real: fun_re229243589l_real ).

tff(sy_c_Fields_Oinverse__class_Oinverse_000tc__Complex__Ocomplex,type,
    invers1449016382omplex: fun_complex_complex ).

tff(sy_c_Fields_Oinverse__class_Oinverse_000tc__Rat__Orat,type,
    inverse_inverse_rat: fun_rat_rat ).

tff(sy_c_Fields_Oinverse__class_Oinverse_000tc__RealDef__Oreal,type,
    inverse_inverse_real: fun_real_real ).

tff(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
    finite_card_int: fun_fun_int_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
    finite_card_nat: fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1876863882t_bool: fun_fu236049833ol_nat ).

tff(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
    finite_finite_int: fun_fu819804446l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
    finite_finite_nat: fun_fu425979586l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    finite1395289673t_bool: fun_fu1540142484l_bool ).

tff(sy_c_FunDef_Opair__leq,type,
    pair_leq: fun_Pr587612383t_bool ).

tff(sy_c_FunDef_Opair__less,type,
    pair_less: fun_Pr587612383t_bool ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod_int_int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_000tc__Int__Oint,type,
    gcd_gcd_int: fun_int_fun_int_int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_000tc__Nat__Onat,type,
    gcd_gcd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
    abs_abs_int: fun_int_int ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__Rat__Orat,type,
    abs_abs_rat: fun_rat_rat ).

tff(sy_c_Groups_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
    abs_abs_real: fun_real_real ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Code____Numeral__Ocode____numeral,type,
    minus_1690775515umeral: fun_Co1323339824umeral ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Complex__Ocomplex,type,
    minus_minus_complex: fun_co822197010omplex ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
    minus_minus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Quickcheck____Narrowing__Ocode____int,type,
    minus_534354567de_int: fun_Qu64793220de_int ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Rat__Orat,type,
    minus_minus_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
    minus_minus_real: fun_re229243589l_real ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Code____Numeral__Ocode____numeral,type,
    one_on1645066479umeral: code_code_numeral ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Complex__Ocomplex,type,
    one_one_complex: complex ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
    one_one_int: int ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Quickcheck____Narrowing__Ocode____int,type,
    one_on1684967323de_int: quickcheck_code_int ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Rat__Orat,type,
    one_one_rat: rat ).

tff(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
    one_one_real: real ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Code____Numeral__Ocode____numeral,type,
    plus_p1627245867umeral: fun_Co1323339824umeral ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Complex__Ocomplex,type,
    plus_plus_complex: fun_co822197010omplex ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
    plus_plus_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Quickcheck____Narrowing__Ocode____int,type,
    plus_p1446045655de_int: fun_Qu64793220de_int ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Rat__Orat,type,
    plus_plus_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
    plus_plus_real: fun_re229243589l_real ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__Complex__Ocomplex,type,
    sgn_sgn_complex: fun_complex_complex ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__Int__Oint,type,
    sgn_sgn_int: fun_int_int ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__Rat__Orat,type,
    sgn_sgn_rat: fun_rat_rat ).

tff(sy_c_Groups_Osgn__class_Osgn_000tc__RealDef__Oreal,type,
    sgn_sgn_real: fun_real_real ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Code____Numeral__Ocode____numeral,type,
    times_1655362735umeral: fun_Co1323339824umeral ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Complex__Ocomplex,type,
    times_times_complex: fun_co822197010omplex ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
    times_times_int: fun_int_fun_int_int ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Quickcheck____Narrowing__Ocode____int,type,
    times_123202395de_int: fun_Qu64793220de_int ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Rat__Orat,type,
    times_times_rat: fun_rat_fun_rat_rat ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
    times_times_real: fun_re229243589l_real ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Complex__Ocomplex,type,
    uminus473333897omplex: fun_complex_complex ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Int__Oint,type,
    uminus_uminus_int: fun_int_int ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__Rat__Orat,type,
    uminus_uminus_rat: fun_rat_rat ).

tff(sy_c_Groups_Ouminus__class_Ouminus_000tc__RealDef__Oreal,type,
    uminus_uminus_real: fun_real_real ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Code____Numeral__Ocode____numeral,type,
    zero_z126310315umeral: code_code_numeral ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Complex__Ocomplex,type,
    zero_zero_complex: complex ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
    zero_zero_int: int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
    zero_zero_nat: nat ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Quickcheck____Narrowing__Ocode____int,type,
    zero_z891286103de_int: quickcheck_code_int ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Rat__Orat,type,
    zero_zero_rat: rat ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
    zero_zero_real: real ).

tff(sy_c_HOL_OThe_000tc__Int__Oint,type,
    the_int: fun_int_bool > int ).

tff(sy_c_HOL_OThe_000tc__RealDef__Oreal,type,
    the_real: fun_real_bool > real ).

tff(sy_c_HOL_OThe_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    the_Pr2103884470nt_int: fun_Pr974702441t_bool > product_prod_int_int ).

tff(sy_c_HOL_OThe_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    the_Pr588456374at_nat: fun_Pr1927655017t_bool > product_prod_nat_nat ).

tff(sy_c_Hilbert__Choice_OEps_000tc__Int__Oint,type,
    hilbert_Eps_int: fun_int_bool > int ).

tff(sy_c_Hilbert__Choice_OEps_000tc__RealDef__Oreal,type,
    hilbert_Eps_real: fun_real_bool > real ).

tff(sy_c_If_000tc__Int__Oint,type,
    if_int: fun_bo1947501544nt_int ).

tff(sy_c_If_000tc__Nat__Onat,type,
    if_nat: fun_bo1071572548at_nat ).

tff(sy_c_If_000tc__RealDef__Oreal,type,
    if_real: fun_bo1998411783l_real ).

tff(sy_c_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    if_Pro1731782967nt_int: fun_bo1552468991nt_int ).

tff(sy_c_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
    if_Pro313124157l_real: fun_bo1189210245l_real ).

tff(sy_c_Int2_OMultInv,type,
    multInv: int > fun_int_int ).

tff(sy_c_IntFact_Od22set,type,
    d22set: fun_int_fun_int_bool ).

tff(sy_c_IntFact_Ozfact,type,
    zfact: fun_int_int ).

tff(sy_c_IntPrimes_Oxzgcd,type,
    xzgcd: ( int * int ) > produc393999548nt_int ).

tff(sy_c_IntPrimes_Oxzgcda,type,
    xzgcda: ( int * int * int * int * int * int * int * int ) > produc393999548nt_int ).

tff(sy_c_IntPrimes_Ozcong,type,
    zcong: int > fun_int_fun_int_bool ).

tff(sy_c_IntPrimes_Ozprime,type,
    zprime: fun_int_bool ).

tff(sy_c_Int_OBit0,type,
    bit0: fun_int_int ).

tff(sy_c_Int_OBit1,type,
    bit1: fun_int_int ).

tff(sy_c_Int_OMin,type,
    min: int ).

tff(sy_c_Int_OPls,type,
    pls: int ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > fun_Pr974702441t_bool ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > fun_Pr974702441t_bool ).

tff(sy_c_Int_Oiszero_000tc__Int__Oint,type,
    iszero_int: fun_int_bool ).

tff(sy_c_Int_Oiszero_000tc__Rat__Orat,type,
    iszero_rat: fun_rat_bool ).

tff(sy_c_Int_Onat,type,
    nat_1: fun_int_nat ).

tff(sy_c_Int_Onat__aux,type,
    nat_aux: int > fun_nat_nat ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Code____Numeral__Ocode____numeral,type,
    number1443263063umeral: fun_in1022115617umeral ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Complex__Ocomplex,type,
    number528085621omplex: fun_int_complex ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
    number_number_of_int: fun_int_int ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
    number_number_of_nat: fun_int_nat ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Quickcheck____Narrowing__Ocode____int,type,
    number1226105091de_int: fun_in1505040589de_int ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Rat__Orat,type,
    number_number_of_rat: fun_int_rat ).

tff(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
    number267125858f_real: fun_int_real ).

tff(sy_c_Int_Opred,type,
    pred: fun_int_int ).

tff(sy_c_Int_Oring__1__class_OInts_000tc__RealDef__Oreal,type,
    ring_1_Ints_real: fun_real_bool ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__Complex__Ocomplex,type,
    ring_11397209091omplex: fun_int_complex ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__Int__Oint,type,
    ring_1_of_int_int: fun_int_int ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__Rat__Orat,type,
    ring_1_of_int_rat: fun_int_rat ).

tff(sy_c_Int_Oring__1__class_Oof__int_000tc__RealDef__Oreal,type,
    ring_1_of_int_real: fun_int_real ).

tff(sy_c_Int_Osucc,type,
    succ: fun_int_int ).

tff(sy_c_Lazy__Sequence_Osmall__lazy_H__rel,type,
    lazy_small_lazy_rel: fun_Pr313471731t_bool ).

tff(sy_c_Legacy__GCD_Ozgcd,type,
    legacy_zgcd: int > fun_int_int ).

tff(sy_c_Lim_OisCont_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
    isCont156215680omplex: fun_complex_complex > fun_complex_bool ).

tff(sy_c_Lim_OisCont_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
    isCont_complex_real: fun_complex_real > fun_complex_bool ).

tff(sy_c_Lim_OisCont_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    isCont_real_real: fun_real_real > fun_real_bool ).

tff(sy_c_Limits_Oat_000tc__Complex__Ocomplex,type,
    at_complex: complex > filter_complex ).

tff(sy_c_Limits_Oat_000tc__RealDef__Oreal,type,
    at_real: real > filter_real ).

tff(sy_c_Limits_Osequentially,type,
    sequentially: filter_nat ).

tff(sy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
    tendst1507391555omplex: ( fun_complex_complex * complex * filter_complex ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
    tendsto_complex_real: ( fun_complex_real * real * filter_complex ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex,type,
    tendsto_nat_complex: ( fun_nat_complex * complex * filter_nat ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal,type,
    tendsto_nat_real: ( fun_nat_real * real * filter_nat ) > bool ).

tff(sy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    tendsto_real_real: ( fun_real_real * real * filter_real ) > bool ).

tff(sy_c_Limits_Otrivial__limit_000tc__Nat__Onat,type,
    trivial_limit_nat: filter_nat > bool ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr313471731t_bool ).

tff(sy_c_Log_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Log_Opowr,type,
    powr: fun_re229243589l_real ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Onat__case_000tc__HOL__Obool,type,
    nat_case_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
    nat_case_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Onat__size,type,
    nat_size: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Code____Numeral__Ocode____numeral,type,
    semiri1619134803umeral: fun_na1360402045umeral ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Complex__Ocomplex,type,
    semiri2020571505omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
    semiri1621563631at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Nat__Onat,type,
    semiri984289939at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Quickcheck____Narrowing__Ocode____i,type,
    semiri1424489471de_int: fun_na295310377de_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Rat__Orat,type,
    semiri151668891at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__RealDef__Oreal,type,
    semiri132038758t_real: fun_nat_real ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Code____Numeral__Ocode____numeral,type,
    size_s945831648umeral: fun_Co116562173al_nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Complex__Ocomplex,type,
    size_size_complex: fun_complex_nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Int__Oint_J,type,
    size_size_list_int: fun_list_int_nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Nat__Onat,type,
    size_size_nat: fun_nat_nat ).

tff(sy_c_Nat__Numeral_Oneg,type,
    nat_neg: fun_int_bool ).

tff(sy_c_Nat__Transfer_Ois__nat,type,
    nat_is_nat: fun_int_bool ).

tff(sy_c_Nat__Transfer_Onat__set,type,
    nat_nat_set: fun_fu819804446l_bool ).

tff(sy_c_Nat__Transfer_Otransfer__morphism_000tc__Int__Oint_000tc__Nat__Onat,type,
    nat_tr876908586nt_nat: fun_int_nat > fun_fu819804446l_bool ).

tff(sy_c_Nat__Transfer_Otransfer__morphism_000tc__Nat__Onat_000tc__Int__Oint,type,
    nat_tr160667106at_int: fun_nat_int > fun_fu425979586l_bool ).

tff(sy_c_Nat__Transfer_Otsub,type,
    nat_tsub: int > fun_int_int ).

tff(sy_c_Nitpick_OFrac,type,
    frac: fun_Pr974702441t_bool ).

tff(sy_c_Nitpick_Oint__gcd,type,
    int_gcd: fun_int_fun_int_int ).

tff(sy_c_Nitpick_Oint__lcm,type,
    int_lcm: int > fun_int_int ).

tff(sy_c_Nitpick_Onat__gcd,type,
    nat_gcd: nat > fun_nat_nat ).

tff(sy_c_Nitpick_Onat__gcd__rel,type,
    nat_gcd_rel: fun_Pr465597171t_bool ).

tff(sy_c_Nitpick_Onat__lcm,type,
    nat_lcm: nat > fun_nat_nat ).

tff(sy_c_Nitpick_Onorm__frac,type,
    norm_frac: ( int * int ) > product_prod_int_int ).

tff(sy_c_Nitpick_Onorm__frac__rel,type,
    norm_frac_rel: fun_Pr313471731t_bool ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na182379510l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Code____Numeral__Ocode____numeral,type,
    ord_le1304079648umeral: fun_Co1372573603l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Quickcheck____Narrowing__Ocode____int,type,
    ord_le1860547276de_int: fun_Qu1069098747t_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
    ord_less_real: fun_re413263731l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le1912455174t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le382113706t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Code____Numeral__Ocode____numeral,type,
    ord_le565307924umeral: fun_Co1372573603l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Quickcheck____Narrowing__Ocode____in,type,
    ord_le258702272de_int: fun_Qu1069098747t_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
    ord_less_eq_real: fun_re413263731l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    ord_le951220754t_bool: fun_fu1151772339l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le1568362934t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Parity_Oeven__odd__class_Oeven_000tc__Int__Oint,type,
    even_odd_even_int: fun_int_bool ).

tff(sy_c_Parity_Oeven__odd__class_Oeven_000tc__Nat__Onat,type,
    even_odd_even_nat: fun_nat_bool ).

tff(sy_c_Power_Opower__class_Opower_000tc__Code____Numeral__Ocode____numeral,type,
    power_2100829034umeral: fun_Co1868595440umeral ).

tff(sy_c_Power_Opower__class_Opower_000tc__Complex__Ocomplex,type,
    power_power_complex: fun_co777804144omplex ).

tff(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_000tc__Quickcheck____Narrowing__Ocode____int,type,
    power_881366806de_int: fun_Qu1439435760de_int ).

tff(sy_c_Power_Opower__class_Opower_000tc__Rat__Orat,type,
    power_power_rat: fun_rat_fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
    power_power_real: fun_re1927296998t_real ).

tff(sy_c_Primes_Ocoprime,type,
    coprime: fun_nat_fun_nat_bool ).

tff(sy_c_Primes_Ofact,type,
    fact: fun_nat_nat ).

tff(sy_c_Primes_Oprime,type,
    prime: fun_nat_bool ).

tff(sy_c_Product__Type_OPair_000tc__Code____Numeral__Ocode____numeral_000tc__Code___,type,
    produc2136830103umeral: fun_Co1101317613umeral ).

tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
    product_Pair_int_int: fun_in1111845413nt_int ).

tff(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__O,type,
    produc282740534nt_int: fun_in1560792517nt_int ).

tff(sy_c_Product__Type_OPair_000tc__Nat__Onat_000tc__Nat__Onat,type,
    product_Pair_nat_nat: fun_na257798253at_nat ).

tff(sy_c_Product__Type_OPair_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Qui,type,
    produc1318306967de_int: fun_Qu1353705541de_int ).

tff(sy_c_Product__Type_OPair_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc865579683l_real: fun_re1550493891l_real ).

tff(sy_c_Product__Type_OPair_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_,type,
    produc713279741t_bool: fun_fu497932381t_bool ).

tff(sy_c_Product__Type_OPair_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod,type,
    produc883642259nt_int: fun_Pr668393395nt_int ).

tff(sy_c_Product__Type_OPair_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod,type,
    produc494345619at_nat: fun_Pr1071803827at_nat ).

tff(sy_c_Product__Type_Oapsnd_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint,type,
    produc713050258nt_int: fun_int_int > fun_Pr241715507nt_int ).

tff(sy_c_Product__Type_Ofst_000tc__Int__Oint_000tc__Int__Oint,type,
    product_fst_int_int: fun_Pr480351984nt_int ).

tff(sy_c_Product__Type_Ofst_000tc__Nat__Onat_000tc__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr1434736532at_nat ).

tff(sy_c_Product__Type_Ofst_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc1935615926l_real: fun_Pr608224961l_real ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__HO,type,
    produc93441119t_bool: fun_int_fun_int_bool > fun_Pr974702441t_bool ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__In,type,
    produc1298267108nt_int: fun_int_fun_int_int > fun_Pr480351984nt_int ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__pr,type,
    produc1518849193nt_int: fun_in1111845413nt_int > fun_Pr241715507nt_int ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HO,type,
    produc1922581855t_bool: fun_nat_fun_nat_bool > fun_Pr1927655017t_bool ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__Nat__Onat_000tc__Nat__Onat_000tc__pr,type,
    produc1391996073at_nat: fun_na257798253at_nat > fun_Pr426809139at_nat ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc595218619l_real: fun_re1550493891l_real > fun_Pr1375198143l_real ).

tff(sy_c_Product__Type_Oprod_Oprod__case_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    produc973645403t_bool: fun_Pr313471731t_bool > fun_Pr1333588703t_bool ).

tff(sy_c_Product__Type_Osnd_000tc__Int__Oint_000tc__Int__Oint,type,
    product_snd_int_int: fun_Pr480351984nt_int ).

tff(sy_c_Product__Type_Osnd_000tc__Nat__Onat_000tc__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr1434736532at_nat ).

tff(sy_c_Product__Type_Osnd_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
    produc556554744l_real: fun_Pr608224961l_real ).

tff(sy_c_Quickcheck__Narrowing_Oaround__zero,type,
    quickc666637781d_zero: int > list_int ).

tff(sy_c_Quickcheck__Narrowing_Oaround__zero__rel,type,
    quickc1265749348ro_rel: fun_int_fun_int_bool ).

tff(sy_c_Quickcheck__Narrowing_Odiv__mod__code__int,type,
    quickc495462417de_int: ( quickcheck_code_int * quickcheck_code_int ) > produc167071911de_int ).

tff(sy_c_Quickcheck__Narrowing_Oint__of,type,
    quickcheck_int_of: quickcheck_code_int > int ).

tff(sy_c_Quickcheck__Narrowing_Onat__of,type,
    quickcheck_nat_of: quickcheck_code_int > nat ).

tff(sy_c_Quickcheck__Narrowing_Oof__int,type,
    quickcheck_of_int: fun_in1505040589de_int ).

tff(sy_c_RComplete_Onatceiling,type,
    natceiling: real > nat ).

tff(sy_c_RComplete_Onatfloor,type,
    natfloor: real > nat ).

tff(sy_c_Rat_OFract,type,
    fract: fun_int_fun_int_rat ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_000tc__RealDef__Oreal,type,
    field_1210416355s_real: fun_real_bool ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr1333588703t_bool ).

tff(sy_c_RealDef_ORatreal,type,
    ratreal: rat > real ).

tff(sy_c_RealDef_Oreal_000tc__Int__Oint,type,
    real_int: fun_int_real ).

tff(sy_c_RealDef_Oreal_000tc__Nat__Onat,type,
    real_nat: fun_nat_real ).

tff(sy_c_RealDef_Ovanishes,type,
    vanishes: fun_nat_rat > bool ).

tff(sy_c_RealVector_Odist__class_Odist_000tc__Complex__Ocomplex,type,
    dist_dist_complex: complex > fun_complex_real ).

tff(sy_c_RealVector_Odist__class_Odist_000tc__RealDef__Oreal,type,
    dist_dist_real: real > fun_real_real ).

tff(sy_c_RealVector_Onorm__class_Onorm_000tc__Complex__Ocomplex,type,
    norm_norm_complex: fun_complex_real ).

tff(sy_c_RealVector_Onorm__class_Onorm_000tc__RealDef__Oreal,type,
    norm_norm_real: fun_real_real ).

tff(sy_c_RealVector_Oof__real_000tc__Complex__Ocomplex,type,
    of_real_complex: real > complex ).

tff(sy_c_RealVector_OscaleR__class_OscaleR_000tc__Complex__Ocomplex,type,
    scaleR1652505878omplex: real > fun_complex_complex ).

tff(sy_c_RealVector_OscaleR__class_OscaleR_000tc__RealDef__Oreal,type,
    scaleR_scaleR_real: real > fun_real_real ).

tff(sy_c_Residues_OLegendre,type,
    legendre: int > fun_int_int ).

tff(sy_c_Residues_OQuadRes,type,
    quadRes: fun_int_fun_int_bool ).

tff(sy_c_Residues_OResSet,type,
    resSet: int > fun_fu819804446l_bool ).

tff(sy_c_Residues_OSR,type,
    sr: fun_int_fun_int_bool ).

tff(sy_c_Residues_OSRStar,type,
    sRStar: fun_int_fun_int_bool ).

tff(sy_c_Residues_OStandardRes,type,
    standardRes: int > fun_int_int ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Code____Numeral__Ocode____numeral,type,
    dvd_dv174992974umeral: fun_Co1372573603l_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Complex__Ocomplex,type,
    dvd_dvd_complex: fun_co2094889951x_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Quickcheck____Narrowing__Ocode____int,type,
    dvd_dv1760642554de_int: fun_Qu1069098747t_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Rings_Odvd__class_Odvd_000tc__RealDef__Oreal,type,
    dvd_dvd_real: fun_re413263731l_bool ).

tff(sy_c_SEQ_OBseq_000tc__RealDef__Oreal,type,
    bseq_real: fun_nat_real > bool ).

tff(sy_c_SEQ_OCauchy_000tc__Complex__Ocomplex,type,
    cauchy_complex: fun_nat_complex > bool ).

tff(sy_c_SEQ_OCauchy_000tc__RealDef__Oreal,type,
    cauchy_real: fun_nat_real > bool ).

tff(sy_c_SEQ_Omonoseq_000tc__RealDef__Oreal,type,
    monoseq_real: fun_nat_real > bool ).

tff(sy_c_SMT_Oz3div,type,
    z3div: int > fun_int_int ).

tff(sy_c_SMT_Oz3mod,type,
    z3mod: int > fun_int_int ).

tff(sy_c_Series_Osuminf_000tc__Complex__Ocomplex,type,
    suminf_complex: fun_fu1930053706omplex ).

tff(sy_c_Series_Osuminf_000tc__RealDef__Oreal,type,
    suminf_real: fun_fu1122561126l_real ).

tff(sy_c_Series_Osummable_000tc__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > bool ).

tff(sy_c_Series_Osummable_000tc__RealDef__Oreal,type,
    summable_real: fun_nat_real > bool ).

tff(sy_c_Series_Osums_000tc__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_bool ).

tff(sy_c_Series_Osums_000tc__RealDef__Oreal,type,
    sums_real: fun_nat_real > fun_real_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Int__Oint,type,
    ord_at875362053st_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Nat__Onat,type,
    ord_at238088361st_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__RealDef__Oreal,type,
    ord_at1589558736t_real: real > fun_re413263731l_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Int__Oint,type,
    ord_at641636577an_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Nat__Onat,type,
    ord_at4362885an_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__RealDef__Oreal,type,
    ord_at1496968948n_real: real > fun_re413263731l_bool ).

tff(sy_c_SetInterval_Oord__class_OatMost_000tc__Nat__Onat,type,
    ord_atMost_nat: fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Int__Oint,type,
    ord_gr1297742076an_int: int > fun_int_fun_int_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Nat__Onat,type,
    ord_gr660468384an_nat: nat > fun_nat_fun_nat_bool ).

tff(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__RealDef__Oreal,type,
    ord_gr788844697n_real: real > fun_re413263731l_bool ).

tff(sy_c_SetInterval_Oord__class_OlessThan_000tc__Nat__Onat,type,
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tff(sy_c_SetInterval_Oord__class_OlessThan_000tc__RealDef__Oreal,type,
    ord_lessThan_real: fun_re413263731l_bool ).

tff(sy_c_Set_OCollect_000tc__Int__Oint,type,
    collect_int: fun_int_bool > fun_int_bool ).

tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
    collect_nat: fun_nat_bool > fun_nat_bool ).

tff(sy_c_Set_OCollect_000tc__RealDef__Oreal,type,
    collect_real: fun_real_bool > fun_real_bool ).

tff(sy_c_Set_OCollect_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    collec1347809874nt_int: fun_Pr974702441t_bool > fun_Pr974702441t_bool ).

tff(sy_c_Set_OCollect_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    collec1979865426at_nat: fun_Pr1927655017t_bool > fun_Pr1927655017t_bool ).

tff(sy_c_Set_OCollect_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pro,type,
    collec50511176nt_int: fun_Pr1333588703t_bool > fun_Pr1333588703t_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Int__Oint,type,
    image_int_int: ( fun_int_int * fun_int_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * fun_int_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    image_275383677t_bool: ( fun_int_fun_int_bool * fun_int_bool ) > fun_fu819804446l_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
    image_nat_int: ( fun_nat_int * fun_nat_bool ) > fun_int_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Ocos,type,
    cos: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Odiffs_000tc__RealDef__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_000tc__RealDef__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln,type,
    ln: fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Osin,type,
    sin: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Otan,type,
    tan: fun_real_real ).

tff(sy_c_TwoSquares__Mirabelle__tyfmylkaxk_Ois__sum2sq,type,
    twoSqu1425020248sum2sq: fun_int_bool ).

tff(sy_c_TwoSquares__Mirabelle__tyfmylkaxk_Osum2sq,type,
    twoSqu2000912462sum2sq: fun_Pr480351984nt_int ).

tff(sy_c_Wellfounded_Oaccp_000tc__Int__Oint,type,
    accp_int: fun_int_fun_int_bool > fun_int_bool ).

tff(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    accp_P2006205492nt_int: fun_Pr313471731t_bool > fun_Pr974702441t_bool ).

tff(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: fun_Pr1927655017t_bool ).

tff(sy_c_WilsonRuss_Oinv,type,
    inv: int > fun_int_int ).

tff(sy_c_WilsonRuss_Owset,type,
    wset: int > fun_int_fun_int_bool ).

tff(sy_c_fEx_000tc__Int__Oint,type,
    fEx_int: fun_fu819804446l_bool ).

tff(sy_c_fEx_000tc__Nat__Onat,type,
    fEx_nat: fun_fu425979586l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: fun_bo1549164019l_bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal_000tc__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_000tc__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_000tc__Rat__Orat,type,
    fequal_rat: fun_rat_fun_rat_bool ).

tff(sy_c_fequal_000tc__RealDef__Oreal,type,
    fequal_real: fun_re413263731l_bool ).

tff(sy_c_fimplies,type,
    fimplies: bool > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__Code____Numeral__Ocode,type,
    hAPP_C498520661umeral: ( fun_Co536835517umeral * code_code_numeral ) > code_code_numeral ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__HOL__Obool,type,
    hAPP_C24501650l_bool: ( fun_Co1584449344l_bool * code_code_numeral ) > bool ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__Nat__Onat,type,
    hAPP_C2129356693al_nat: ( fun_Co116562173al_nat * code_code_numeral ) > nat ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Code____Numer,type,
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tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Code____Numer_248,type,
    hAPP_C10902773l_bool: ( fun_Co1372573603l_bool * code_code_numeral ) > fun_Co1584449344l_bool ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Code____Numer_249,type,
    hAPP_C1614127039umeral: ( fun_Co1101317613umeral * code_code_numeral ) > fun_Co991465098umeral ).

tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__fun_Itc__Nat__Onat_Mtc,type,
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tff(sy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__prod_Itc__Code____Nume,type,
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tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
    hAPP_complex_complex: ( fun_complex_complex * complex ) > complex ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__HOL__Obool,type,
    hAPP_complex_bool: ( fun_complex_bool * complex ) > bool ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__Nat__Onat,type,
    hAPP_complex_nat: ( fun_complex_nat * complex ) > nat ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
    hAPP_complex_real: ( fun_complex_real * complex ) > real ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__fun_Itc__Complex__Ocomplex_Mtc__Comple,type,
    hAPP_c172932010omplex: ( fun_co822197010omplex * complex ) > fun_complex_complex ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__fun_Itc__Complex__Ocomplex_Mtc__HOL__O,type,
    hAPP_c1403687985x_bool: ( fun_co2094889951x_bool * complex ) > fun_complex_bool ).

tff(sy_c_hAPP_000tc__Complex__Ocomplex_000tc__fun_Itc__Nat__Onat_Mtc__Complex__Ocomp,type,
    hAPP_c1088319240omplex: ( fun_co777804144omplex * complex ) > fun_nat_complex ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
    hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mt,type,
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tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mt,type,
    hAPP_b992065680at_nat: ( fun_bo1071572548at_nat * bool ) > fun_nat_fun_nat_nat ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef,type,
    hAPP_b646336293l_real: ( fun_bo1998411783l_real * bool ) > fun_re229243589l_real ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    hAPP_b40753821nt_int: ( fun_bo1552468991nt_int * bool ) > fun_Pr1580971581nt_int ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDe,type,
    hAPP_b1482221219l_real: ( fun_bo1189210245l_real * bool ) > fun_Pr1283822147l_real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Code____Numeral__Ocode____numeral,type,
    hAPP_i769753017umeral: ( fun_in1022115617umeral * int ) > code_code_numeral ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Complex__Ocomplex,type,
    hAPP_int_complex: ( fun_int_complex * int ) > complex ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
    hAPP_int_bool: ( fun_int_bool * int ) > bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,type,
    hAPP_int_int: ( fun_int_int * int ) > int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Nat__Onat,type,
    hAPP_int_nat: ( fun_int_nat * int ) > nat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Quickcheck____Narrowing__Ocode____int,type,
    hAPP_i1732201573de_int: ( fun_in1505040589de_int * int ) > quickcheck_code_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__Rat__Orat,type,
    hAPP_int_rat: ( fun_int_rat * int ) > rat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__RealDef__Oreal,type,
    hAPP_int_real: ( fun_int_real * int ) > real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_i68813070l_bool: ( fun_in531499254l_bool * int ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
    hAPP_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J,type,
    hAPP_int_fun_int_rat: ( fun_int_fun_int_rat * int ) > fun_int_rat ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J,type,
    hAPP_i627165055t_real: ( fun_int_fun_int_real * int ) > fun_int_real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mt,type,
    hAPP_i1823918693l_bool: ( fun_in397122125l_bool * int ) > fun_in531499254l_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc,type,
    hAPP_i345030060t_bool: ( fun_in351380826t_bool * int ) > fun_int_fun_int_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc_250,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int,type,
    hAPP_i1153444985nt_int: ( fun_in2124216673nt_int * int ) > fun_in989261322nt_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mt,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    hAPP_i418383825t_bool: ( fun_int_fun_nat_bool * int ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
    hAPP_int_fun_nat_int: ( fun_int_fun_nat_int * int ) > fun_nat_int ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J,type,
    hAPP_i157317923t_real: ( fun_int_fun_nat_real * int ) > fun_nat_real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mt,type,
    hAPP_i614188481l_bool: ( fun_in1678813097l_bool * int ) > fun_na1469252690l_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Rat__Orat_Mtc__HOL__Obool_J,type,
    hAPP_i16292313t_bool: ( fun_int_fun_rat_bool * int ) > fun_rat_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
    hAPP_i371647666l_real: ( fun_in7315610l_real * int ) > fun_real_real ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
    hAPP_i2112223885l_bool: ( fun_in938817141l_bool * int ) > fun_fu819804446l_bool ).

tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J__251,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_M,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_M_252,type,
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tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Orea,type,
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    hAPP_i119638232t_bool: ( fun_in1698189120t_bool * int ) > fun_Pr974702441t_bool ).

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    hAPP_i61245889nt_int: ( fun_in1211676399nt_int * int ) > fun_Pr480351984nt_int ).

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    hAPP_i2104874254nt_int: ( fun_in941977532nt_int * int ) > fun_Pr1580971581nt_int ).

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    hAPP_i1216375074nt_int: ( fun_in989261322nt_int * int ) > fun_Pr241715507nt_int ).

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    hAPP_i1730167831nt_int: ( fun_in1560792517nt_int * int ) > fun_Pr1076553414nt_int ).

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    hAPP_i1524277240nt_int: ( fun_in2003698982nt_int * int ) > product_prod_int_int ).

tff(sy_c_hAPP_000tc__List__Olist_Itc__Int__Oint_J_000tc__Nat__Onat,type,
    hAPP_list_int_nat: ( fun_list_int_nat * list_int ) > nat ).

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    hAPP_n1108039445umeral: ( fun_na1360402045umeral * nat ) > code_code_numeral ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Complex__Ocomplex,type,
    hAPP_nat_complex: ( fun_nat_complex * nat ) > complex ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
    hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
    hAPP_nat_int: ( fun_nat_int * nat ) > int ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
    hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).

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    hAPP_n522471361de_int: ( fun_na295310377de_int * nat ) > quickcheck_code_int ).

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    hAPP_nat_rat: ( fun_nat_rat * nat ) > rat ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__RealDef__Oreal,type,
    hAPP_nat_real: ( fun_nat_real * nat ) > real ).

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    hAPP_n1006566506l_bool: ( fun_na1469252690l_bool * nat ) > fun_bool_bool ).

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    hAPP_nat_fun_int_int: ( fun_nat_fun_int_int * nat ) > fun_int_int ).

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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__RealDef__Ore,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__Nat,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__RealDef_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__415,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__416,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc__fun_Itc__417,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_Mtc__fun_Itc,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_Mtc__fun_Itc_418,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_Mtc__fun_Itc_419,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_420,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_421,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__fun_422,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_M,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_M_423,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__RealDe,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__RealDe_424,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Oreal_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Oreal__425,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mtc_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mtc_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun,type,
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pro,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__HOL__Obool_J_426,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J__427,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J__428,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J__429,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J__430,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J__431,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Int__Oint_J__432,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__Rat__Orat_J_,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__HOL,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__HOL_433,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_434,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_435,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_436,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_437,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_438,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__Int_439,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_440,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__fun_441,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro_442,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro_443,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__fun_Itc__pro_444,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__In,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__HO,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_Mtc__Re,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fu,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pr,type,
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tff(sy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__pr,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Rat__Orat,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__HOL__Obool,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint_,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint__445,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint__446,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__Int__Oint__447,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p_448,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__fun_Itc__p_449,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc__,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___450,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___451,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___452,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__prod_Itc___453,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc__Int__Oint,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc__Int__Oint_454,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc__prod_Itc_,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__HOL__Obool,type,
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tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__Nat__Onat,type,
    hAPP_P1654798560at_nat: ( fun_Pr1434736532at_nat * product_prod_nat_nat ) > nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__fun_Itc__prod_Itc__,type,
    hAPP_P402336767at_nat: ( fun_Pr1071803827at_nat * product_prod_nat_nat ) > fun_Pr1858073897at_nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_Itc__Nat__Onat,type,
    hAPP_P1586233937at_nat: ( fun_Pr426809139at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_Itc__prod_Itc_,type,
    hAPP_P369611207at_nat: ( fun_Pr1858073897at_nat * product_prod_nat_nat ) > produc1322466333at_nat ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__HOL__Oboo,type,
    hAPP_P1333854989l_bool: ( fun_Pr1127657839l_bool * produc914805421l_real ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__RealDef__,type,
    hAPP_P731461727l_real: ( fun_Pr608224961l_real * produc914805421l_real ) > real ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__fun_Itc__,type,
    hAPP_P1982799375l_real: ( fun_Pr1283822147l_real * produc914805421l_real ) > fun_Pr1375198143l_real ).

tff(sy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__prod_Itc_,type,
    hAPP_P1860904029l_real: ( fun_Pr1375198143l_real * produc914805421l_real ) > produc914805421l_real ).

tff(sy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__In,type,
    hAPP_P880235623l_bool: ( fun_Pr4007113l_bool * produc1450482951t_bool ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__In_455,type,
    hAPP_P1681069257l_bool: ( fun_Pr1669804459l_bool * produc1450482951t_bool ) > fun_fu2145628351l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__I,type,
    hAPP_P62333565t_bool: ( fun_Pr1333588703t_bool * produc1137372701nt_int ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__I_456,type,
    hAPP_P1062890741l_bool: ( fun_Pr1020157655l_bool * produc1137372701nt_int ) > fun_fu1202355157l_bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__N,type,
    hAPP_P1463840893t_bool: ( fun_Pr587612383t_bool * produc1322466333at_nat ) > bool ).

tff(sy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__N_457,type,
    hAPP_P1876494581l_bool: ( fun_Pr822475479l_bool * produc1322466333at_nat ) > fun_fu791988693l_bool ).

tff(sy_c_hBOOL,type,
    hBOOL: bool > $o ).

tff(sy_c_member_000tc__Int__Oint,type,
    member_int: fun_in938817141l_bool ).

tff(sy_c_member_000tc__Nat__Onat,type,
    member_nat: fun_na1436237685l_bool ).

tff(sy_c_member_000tc__RealDef__Oreal,type,
    member_real: fun_re1865242107l_bool ).

tff(sy_c_member_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
    member_fun_int_bool: fun_fu1637550549l_bool ).

tff(sy_c_member_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__,type,
    member308914164t_bool: fun_Pr1669804459l_bool ).

tff(sy_c_member_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc_,type,
    member2143287562nt_int: fun_Pr1020157655l_bool ).

tff(sy_c_member_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc_,type,
    member180897546at_nat: fun_Pr822475479l_bool ).

tff(sy_v_m,type,
    m: int ).

tff(sy_v_s1____,type,
    s1: int ).

tff(sy_v_s____,type,
    s: int ).

tff(sy_v_t____,type,
    t: int ).

%----Relevant facts (5196)
tff(fact_0__096t_A_060_A0_096,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),zero_zero_int)) ).

tff(fact_1_calculation_I1_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),one_one_int)) ).

tff(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),zero_zero_int))) ).

tff(fact_3_t,axiom,
    hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t) ).

tff(fact_4_calculation_I2_J,axiom,
    ( ( t = zero_zero_int )
   => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) = zero_zero_int ) ) ).

tff(fact_5__096_126_A1_A_060_061_At_096,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) ).

tff(fact_6_p0,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_7_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int)) ).

tff(fact_8_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) ).

tff(fact_9_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat)) ).

tff(fact_10_sum__power2__gt__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
    <=> ( ( X_1 != zero_zero_int )
        | ( Y_2 != zero_zero_int ) ) ) ).

tff(fact_11_sum__power2__gt__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
    <=> ( ( X_1 != zero_zero_real )
        | ( Y_2 != zero_zero_real ) ) ) ).

tff(fact_12_sum__power2__gt__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
    <=> ( ( X_1 != zero_zero_rat )
        | ( Y_2 != zero_zero_rat ) ) ) ).

tff(fact_13_sum__power2__eq__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_int )
    <=> ( ( X_1 = zero_zero_int )
        & ( Y_2 = zero_zero_int ) ) ) ).

tff(fact_14_sum__power2__eq__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_15_sum__power2__eq__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_rat )
    <=> ( ( X_1 = zero_zero_rat )
        & ( Y_2 = zero_zero_rat ) ) ) ).

tff(fact_16_power2__less__0,axiom,
    ! [A_240: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_240),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_int)) ).

tff(fact_17_power2__less__0,axiom,
    ! [A_240: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_240),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).

tff(fact_18_power2__less__0,axiom,
    ! [A_240: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_240),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_rat)) ).

tff(fact_19_zero__less__power2,axiom,
    ! [A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
    <=> ( A != zero_zero_int ) ) ).

tff(fact_20_zero__less__power2,axiom,
    ! [A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
    <=> ( A != zero_zero_real ) ) ).

tff(fact_21_zero__less__power2,axiom,
    ! [A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
    <=> ( A != zero_zero_rat ) ) ).

tff(fact_22_one__power2,axiom,
    hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int ).

tff(fact_23_one__power2,axiom,
    hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_nat ).

tff(fact_24_one__power2,axiom,
    hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_real ).

tff(fact_25_one__power2,axiom,
    hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_on1645066479umeral ).

tff(fact_26_one__power2,axiom,
    hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_complex ).

tff(fact_27_one__power2,axiom,
    hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_on1684967323de_int ).

tff(fact_28_one__power2,axiom,
    hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_rat ).

tff(fact_29_zero__power2,axiom,
    hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ).

tff(fact_30_zero__power2,axiom,
    hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_nat ).

tff(fact_31_zero__power2,axiom,
    hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_real ).

tff(fact_32_zero__power2,axiom,
    hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral ).

tff(fact_33_zero__power2,axiom,
    hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex ).

tff(fact_34_zero__power2,axiom,
    hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int ).

tff(fact_35_zero__power2,axiom,
    hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_rat ).

tff(fact_36_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

tff(fact_37_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

tff(fact_38_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

tff(fact_39_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

tff(fact_40_add__special_I2_J,axiom,
    ! [W_17: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_17)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17)) ).

tff(fact_41_add__special_I2_J,axiom,
    ! [W_17: int] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_17)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17)) ).

tff(fact_42_add__special_I2_J,axiom,
    ! [W_17: int] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_17)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17)) ).

tff(fact_43_add__special_I2_J,axiom,
    ! [W_17: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_17)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17)) ).

tff(fact_44_add__special_I3_J,axiom,
    ! [V_23: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_23)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).

tff(fact_45_add__special_I3_J,axiom,
    ! [V_23: int] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_23)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).

tff(fact_46_add__special_I3_J,axiom,
    ! [V_23: int] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_23)),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).

tff(fact_47_add__special_I3_J,axiom,
    ! [V_23: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_23)),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) ).

tff(fact_48_t__l__p,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_49__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
    ~ ! [T: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),T) ).

tff(fact_50_p,axiom,
    hBOOL(hAPP_int_bool(zprime,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_51_qf1pt,axiom,
    hBOOL(hAPP_int_bool(twoSqu1425020248sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t))) ).

tff(fact_52_zle__refl,axiom,
    ! [W: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).

tff(fact_53_number__of__is__id,axiom,
    ! [K_2: int] : hAPP_int_int(number_number_of_int,K_2) = K_2 ).

tff(fact_54_zmult__commute,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1) ).

tff(fact_55_zle__linear,axiom,
    ! [Z_1: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) ) ).

tff(fact_56_times__numeral__code_I5_J,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),W)) ).

tff(fact_57_less__eq__number__of__int__code,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_58_le__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_59_le__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_60_le__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_61_zmult__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ).

tff(fact_62_zle__trans,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K_2)) ) ) ).

tff(fact_63_zle__antisym,axiom,
    ! [Z_1: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1))
       => ( Z_1 = W ) ) ) ).

tff(fact_64_zpower__zadd__distrib,axiom,
    ! [X: int,Y: nat,Z_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)) ).

tff(fact_65_less__eq__int__code_I16_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_66_rel__simps_I34_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_67_rel__simps_I19_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).

tff(fact_68_less__eq__int__code_I13_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_69_rel__simps_I31_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_70_zless__le,axiom,
    ! [Z_2: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1))
        & ( Z_2 != W_1 ) ) ) ).

tff(fact_71_zadd__left__mono,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),J_2))) ) ).

tff(fact_72_eq__number__of__0,axiom,
    ! [V: int] :
      ( ( hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ).

tff(fact_73_eq__0__number__of,axiom,
    ! [V: int] :
      ( ( zero_zero_nat = hAPP_int_nat(number_number_of_nat,V) )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ).

tff(fact_74_semiring__mult__number__of,axiom,
    ! [V_22: int,V_21: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
       => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_21)),hAPP_int_int(number_number_of_int,V_22)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ) ).

tff(fact_75_semiring__mult__number__of,axiom,
    ! [V_22: int,V_21: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_21)),hAPP_int_nat(number_number_of_nat,V_22)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ) ).

tff(fact_76_semiring__mult__number__of,axiom,
    ! [V_22: int,V_21: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
       => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_21)),hAPP_int_real(number267125858f_real,V_22)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ) ).

tff(fact_77_semiring__mult__number__of,axiom,
    ! [V_22: int,V_21: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
       => ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_21)),hAPP_int_complex(number528085621omplex,V_22)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ) ).

tff(fact_78_semiring__mult__number__of,axiom,
    ! [V_22: int,V_21: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22))
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_21)),hAPP_int_rat(number_number_of_rat,V_22)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) ) ) ) ).

tff(fact_79_mult__number__of__left,axiom,
    ! [V_20: int,W_16: int,Z_15: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_20)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_16)),Z_15)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15) ).

tff(fact_80_mult__number__of__left,axiom,
    ! [V_20: int,W_16: int,Z_15: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_20)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_16)),Z_15)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15) ).

tff(fact_81_mult__number__of__left,axiom,
    ! [V_20: int,W_16: int,Z_15: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_20)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_16)),Z_15)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15) ).

tff(fact_82_mult__number__of__left,axiom,
    ! [V_20: int,W_16: int,Z_15: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_20)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_16)),Z_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15) ).

tff(fact_83_arith__simps_I32_J,axiom,
    ! [V_19: int,W_15: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_19)),hAPP_int_int(number_number_of_int,W_15)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15)) ).

tff(fact_84_arith__simps_I32_J,axiom,
    ! [V_19: int,W_15: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_19)),hAPP_int_real(number267125858f_real,W_15)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15)) ).

tff(fact_85_arith__simps_I32_J,axiom,
    ! [V_19: int,W_15: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_19)),hAPP_int_complex(number528085621omplex,W_15)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15)) ).

tff(fact_86_arith__simps_I32_J,axiom,
    ! [V_19: int,W_15: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_19)),hAPP_int_rat(number_number_of_rat,W_15)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15)) ).

tff(fact_87_number__of__mult,axiom,
    ! [V_18: int,W_14: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_18)),hAPP_int_int(number_number_of_int,W_14)) ).

tff(fact_88_number__of__mult,axiom,
    ! [V_18: int,W_14: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_18)),hAPP_int_real(number267125858f_real,W_14)) ).

tff(fact_89_number__of__mult,axiom,
    ! [V_18: int,W_14: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_18)),hAPP_int_complex(number528085621omplex,W_14)) ).

tff(fact_90_number__of__mult,axiom,
    ! [V_18: int,W_14: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_18)),hAPP_int_rat(number_number_of_rat,W_14)) ).

tff(fact_91_sum__squares__le__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))),zero_zero_int))
    <=> ( ( X_1 = zero_zero_int )
        & ( Y_2 = zero_zero_int ) ) ) ).

tff(fact_92_sum__squares__le__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))),zero_zero_real))
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_93_sum__squares__le__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))),zero_zero_rat))
    <=> ( ( X_1 = zero_zero_rat )
        & ( Y_2 = zero_zero_rat ) ) ) ).

tff(fact_94_sum__squares__ge__zero,axiom,
    ! [X_47: int,Y_35: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_47),X_47)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_35),Y_35)))) ).

tff(fact_95_sum__squares__ge__zero,axiom,
    ! [X_47: real,Y_35: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_47),X_47)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_35),Y_35)))) ).

tff(fact_96_sum__squares__ge__zero,axiom,
    ! [X_47: rat,Y_35: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_47),X_47)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_35),Y_35)))) ).

tff(fact_97_le__special_I3_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),pls)) ) ).

tff(fact_98_le__special_I3_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),zero_zero_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),pls)) ) ).

tff(fact_99_le__special_I3_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),zero_zero_rat))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),pls)) ) ).

tff(fact_100_le__special_I1_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).

tff(fact_101_le__special_I1_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).

tff(fact_102_le__special_I1_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) ) ).

tff(fact_103_less__0__number__of,axiom,
    ! [V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)) ) ).

tff(fact_104_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,V)),hAPP_i769753017umeral(number1443263063umeral,W_1)))
    <=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,W_1)),hAPP_i769753017umeral(number1443263063umeral,V))) ) ).

tff(fact_105_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1)))
    <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W_1)),hAPP_int_int(number_number_of_int,V))) ) ).

tff(fact_106_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,W_1)))
    <=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,W_1)),hAPP_int_nat(number_number_of_nat,V))) ) ).

tff(fact_107_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,V)),hAPP_int_real(number267125858f_real,W_1)))
    <=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,W_1)),hAPP_int_real(number267125858f_real,V))) ) ).

tff(fact_108_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,V)),hAPP_i1732201573de_int(number1226105091de_int,W_1)))
    <=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,W_1)),hAPP_i1732201573de_int(number1226105091de_int,V))) ) ).

tff(fact_109_le__number__of__eq__not__less,axiom,
    ! [V: int,W_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,V)),hAPP_int_rat(number_number_of_rat,W_1)))
    <=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,W_1)),hAPP_int_rat(number_number_of_rat,V))) ) ).

tff(fact_110_rel__simps_I22_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).

tff(fact_111_less__eq__int__code_I14_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_112_rel__simps_I32_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_113_rel__simps_I27_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),pls)) ) ).

tff(fact_114_rel__simps_I21_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).

tff(fact_115_zadd__zless__mono,axiom,
    ! [Z_4: int,Z_1: int,W_13: int,W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_13),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_13),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1))) ) ) ).

tff(fact_116_nat__number__of__Pls,axiom,
    hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).

tff(fact_117_semiring__norm_I113_J,axiom,
    zero_zero_nat = hAPP_int_nat(number_number_of_nat,pls) ).

tff(fact_118_le__special_I4_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),one_one_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(bit1,pls))) ) ).

tff(fact_119_le__special_I4_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),one_one_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(bit1,pls))) ) ).

tff(fact_120_le__special_I4_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),one_one_rat))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(bit1,pls))) ) ).

tff(fact_121_le__special_I2_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).

tff(fact_122_le__special_I2_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).

tff(fact_123_le__special_I2_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).

tff(fact_124_nat__1__add__1,axiom,
    hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_125_mult__Pls,axiom,
    ! [W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls ).

tff(fact_126_mult__Bit0,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit0,K_2)),L_2) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2)) ).

tff(fact_127_less__number__of__int__code,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_128_zmult__1__right,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),one_one_int) = Z_1 ).

tff(fact_129_zmult__1,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z_1) = Z_1 ).

tff(fact_130_plus__numeral__code_I9_J,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),W)) ).

tff(fact_131_zadd__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).

tff(fact_132_zadd__zmult__distrib2,axiom,
    ! [W: int,Z1: int,Z2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).

tff(fact_133_rel__simps_I29_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).

tff(fact_134_rel__simps_I5_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) ) ).

tff(fact_135_less__eq__int__code_I15_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_136_rel__simps_I33_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_137_less__int__code_I14_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

tff(fact_138_mem__def,axiom,
    ! [X_1: int,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1))
    <=> hBOOL(hAPP_int_bool(A_1,X_1)) ) ).

tff(fact_139_mem__def,axiom,
    ! [X_1: produc1322466333at_nat,A_1: fun_Pr587612383t_bool] :
      ( hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,X_1),A_1))
    <=> hBOOL(hAPP_P1463840893t_bool(A_1,X_1)) ) ).

tff(fact_140_mem__def,axiom,
    ! [X_1: fun_int_bool,A_1: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A_1))
    <=> hBOOL(hAPP_f448129468l_bool(A_1,X_1)) ) ).

tff(fact_141_mem__def,axiom,
    ! [X_1: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
    <=> hBOOL(hAPP_nat_bool(A_1,X_1)) ) ).

tff(fact_142_mem__def,axiom,
    ! [X_1: real,A_1: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),A_1))
    <=> hBOOL(hAPP_real_bool(A_1,X_1)) ) ).

tff(fact_143_mem__def,axiom,
    ! [X_1: produc1137372701nt_int,A_1: fun_Pr1333588703t_bool] :
      ( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,X_1),A_1))
    <=> hBOOL(hAPP_P62333565t_bool(A_1,X_1)) ) ).

tff(fact_144_mem__def,axiom,
    ! [X_1: produc1450482951t_bool,A_1: fun_Pr4007113l_bool] :
      ( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,X_1),A_1))
    <=> hBOOL(hAPP_P880235623l_bool(A_1,X_1)) ) ).

tff(fact_145_Collect__def,axiom,
    ! [P_1: fun_int_bool] : collect_int(P_1) = P_1 ).

tff(fact_146_Collect__def,axiom,
    ! [P_1: fun_nat_bool] : collect_nat(P_1) = P_1 ).

tff(fact_147_Collect__def,axiom,
    ! [P_1: fun_Pr1333588703t_bool] : collec50511176nt_int(P_1) = P_1 ).

tff(fact_148_Collect__def,axiom,
    ! [P_1: fun_Pr974702441t_bool] : collec1347809874nt_int(P_1) = P_1 ).

tff(fact_149_Collect__def,axiom,
    ! [P_1: fun_real_bool] : collect_real(P_1) = P_1 ).

tff(fact_150_Collect__def,axiom,
    ! [P_1: fun_Pr1927655017t_bool] : collec1979865426at_nat(P_1) = P_1 ).

tff(fact_151_rel__simps_I15_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) ) ).

tff(fact_152_less__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1)) )
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) ) ) ).

tff(fact_153_int__one__le__iff__zero__less,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).

tff(fact_154_nat__numeral__1__eq__1,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).

tff(fact_155_Numeral1__eq1__nat,axiom,
    one_one_nat = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) ).

tff(fact_156_zless__imp__add1__zle,axiom,
    ! [W: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1)) ) ).

tff(fact_157_add1__zle__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

tff(fact_158_zle__add1__eq__le,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ).

tff(fact_159_semiring__add__number__of,axiom,
    ! [V_17: int,V_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
       => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_16)),hAPP_int_int(number_number_of_int,V_17)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) ) ) ) ).

tff(fact_160_semiring__add__number__of,axiom,
    ! [V_17: int,V_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_16)),hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) ) ) ) ).

tff(fact_161_semiring__add__number__of,axiom,
    ! [V_17: int,V_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_16)),hAPP_int_real(number267125858f_real,V_17)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) ) ) ) ).

tff(fact_162_semiring__add__number__of,axiom,
    ! [V_17: int,V_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
       => ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_16)),hAPP_int_complex(number528085621omplex,V_17)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) ) ) ) ).

tff(fact_163_semiring__add__number__of,axiom,
    ! [V_17: int,V_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17))
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_16)),hAPP_int_rat(number_number_of_rat,V_17)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) ) ) ) ).

tff(fact_164_add__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_3) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3)) ) ) ) ) ) ).

tff(fact_165_le__imp__0__less,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1))) ) ).

tff(fact_166_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_int(number_number_of_int,X_1) = hAPP_int_int(number_number_of_int,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_167_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_real(number267125858f_real,X_1) = hAPP_int_real(number267125858f_real,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_168_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_complex(number528085621omplex,X_1) = hAPP_int_complex(number528085621omplex,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_169_eq__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_rat(number_number_of_rat,X_1) = hAPP_int_rat(number_number_of_rat,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_170_number__of__reorient,axiom,
    ! [W_1: int,X_1: nat] :
      ( ( hAPP_int_nat(number_number_of_nat,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_nat(number_number_of_nat,W_1) ) ) ).

tff(fact_171_number__of__reorient,axiom,
    ! [W_1: int,X_1: int] :
      ( ( hAPP_int_int(number_number_of_int,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_int(number_number_of_int,W_1) ) ) ).

tff(fact_172_number__of__reorient,axiom,
    ! [W_1: int,X_1: real] :
      ( ( hAPP_int_real(number267125858f_real,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_real(number267125858f_real,W_1) ) ) ).

tff(fact_173_number__of__reorient,axiom,
    ! [W_1: int,X_1: code_code_numeral] :
      ( ( hAPP_i769753017umeral(number1443263063umeral,W_1) = X_1 )
    <=> ( X_1 = hAPP_i769753017umeral(number1443263063umeral,W_1) ) ) ).

tff(fact_174_number__of__reorient,axiom,
    ! [W_1: int,X_1: complex] :
      ( ( hAPP_int_complex(number528085621omplex,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_complex(number528085621omplex,W_1) ) ) ).

tff(fact_175_number__of__reorient,axiom,
    ! [W_1: int,X_1: quickcheck_code_int] :
      ( ( hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_1 )
    <=> ( X_1 = hAPP_i1732201573de_int(number1226105091de_int,W_1) ) ) ).

tff(fact_176_number__of__reorient,axiom,
    ! [W_1: int,X_1: rat] :
      ( ( hAPP_int_rat(number_number_of_rat,W_1) = X_1 )
    <=> ( X_1 = hAPP_int_rat(number_number_of_rat,W_1) ) ) ).

tff(fact_177_rel__simps_I51_J,axiom,
    ! [K_1: int,L: int] :
      ( ( hAPP_int_int(bit1,K_1) = hAPP_int_int(bit1,L) )
    <=> ( K_1 = L ) ) ).

tff(fact_178_rel__simps_I48_J,axiom,
    ! [K_1: int,L: int] :
      ( ( hAPP_int_int(bit0,K_1) = hAPP_int_int(bit0,L) )
    <=> ( K_1 = L ) ) ).

tff(fact_179_zless__linear,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
      | ( X = Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).

tff(fact_180_sum__squares__eq__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2)) = zero_zero_int )
    <=> ( ( X_1 = zero_zero_int )
        & ( Y_2 = zero_zero_int ) ) ) ).

tff(fact_181_sum__squares__eq__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_182_sum__squares__eq__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)) = zero_zero_rat )
    <=> ( ( X_1 = zero_zero_rat )
        & ( Y_2 = zero_zero_rat ) ) ) ).

tff(fact_183_left__distrib__number__of,axiom,
    ! [A_239: code_code_numeral,B_167: code_code_numeral,V_15: int] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_239),B_167)),hAPP_i769753017umeral(number1443263063umeral,V_15)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_239),hAPP_i769753017umeral(number1443263063umeral,V_15))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_167),hAPP_i769753017umeral(number1443263063umeral,V_15))) ).

tff(fact_184_left__distrib__number__of,axiom,
    ! [A_239: int,B_167: int,V_15: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_239),B_167)),hAPP_int_int(number_number_of_int,V_15)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_239),hAPP_int_int(number_number_of_int,V_15))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_167),hAPP_int_int(number_number_of_int,V_15))) ).

tff(fact_185_left__distrib__number__of,axiom,
    ! [A_239: nat,B_167: nat,V_15: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_239),B_167)),hAPP_int_nat(number_number_of_nat,V_15)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_239),hAPP_int_nat(number_number_of_nat,V_15))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_167),hAPP_int_nat(number_number_of_nat,V_15))) ).

tff(fact_186_left__distrib__number__of,axiom,
    ! [A_239: real,B_167: real,V_15: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_239),B_167)),hAPP_int_real(number267125858f_real,V_15)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_239),hAPP_int_real(number267125858f_real,V_15))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_167),hAPP_int_real(number267125858f_real,V_15))) ).

tff(fact_187_left__distrib__number__of,axiom,
    ! [A_239: complex,B_167: complex,V_15: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_239),B_167)),hAPP_int_complex(number528085621omplex,V_15)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_239),hAPP_int_complex(number528085621omplex,V_15))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_167),hAPP_int_complex(number528085621omplex,V_15))) ).

tff(fact_188_left__distrib__number__of,axiom,
    ! [A_239: quickcheck_code_int,B_167: quickcheck_code_int,V_15: int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_239),B_167)),hAPP_i1732201573de_int(number1226105091de_int,V_15)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_239),hAPP_i1732201573de_int(number1226105091de_int,V_15))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_167),hAPP_i1732201573de_int(number1226105091de_int,V_15))) ).

tff(fact_189_left__distrib__number__of,axiom,
    ! [A_239: rat,B_167: rat,V_15: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_239),B_167)),hAPP_int_rat(number_number_of_rat,V_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_239),hAPP_int_rat(number_number_of_rat,V_15))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_167),hAPP_int_rat(number_number_of_rat,V_15))) ).

tff(fact_190_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: code_code_numeral,C_97: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_14)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_166),C_97)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_14)),B_166)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_14)),C_97)) ).

tff(fact_191_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: int,C_97: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_166),C_97)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),B_166)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),C_97)) ).

tff(fact_192_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: nat,C_97: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_14)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_166),C_97)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_14)),B_166)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_14)),C_97)) ).

tff(fact_193_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: real,C_97: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_166),C_97)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),B_166)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),C_97)) ).

tff(fact_194_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: complex,C_97: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_166),C_97)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),B_166)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),C_97)) ).

tff(fact_195_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: quickcheck_code_int,C_97: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_14)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_166),C_97)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_14)),B_166)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_14)),C_97)) ).

tff(fact_196_right__distrib__number__of,axiom,
    ! [V_14: int,B_166: rat,C_97: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_166),C_97)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),B_166)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),C_97)) ).

tff(fact_197_zadd__assoc,axiom,
    ! [Z1: int,Z2: int,Z3: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ).

tff(fact_198_zadd__left__commute,axiom,
    ! [X: int,Y: int,Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z_1)) ).

tff(fact_199_zadd__commute,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1) ).

tff(fact_200_zero__is__num__zero,axiom,
    zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).

tff(fact_201_zmult__zless__mono2,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2))) ) ) ).

tff(fact_202_power2__eq__imp__eq,axiom,
    ! [X_46: int,Y_34: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_46))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_34))
         => ( X_46 = Y_34 ) ) ) ) ).

tff(fact_203_power2__eq__imp__eq,axiom,
    ! [X_46: nat,Y_34: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_46))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_34))
         => ( X_46 = Y_34 ) ) ) ) ).

tff(fact_204_power2__eq__imp__eq,axiom,
    ! [X_46: real,Y_34: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_46))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_34))
         => ( X_46 = Y_34 ) ) ) ) ).

tff(fact_205_power2__eq__imp__eq,axiom,
    ! [X_46: code_code_numeral,Y_34: code_code_numeral] :
      ( ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_46))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_34))
         => ( X_46 = Y_34 ) ) ) ) ).

tff(fact_206_power2__eq__imp__eq,axiom,
    ! [X_46: quickcheck_code_int,Y_34: quickcheck_code_int] :
      ( ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_46))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_34))
         => ( X_46 = Y_34 ) ) ) ) ).

tff(fact_207_power2__eq__imp__eq,axiom,
    ! [X_46: rat,Y_34: rat] :
      ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_46))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_34))
         => ( X_46 = Y_34 ) ) ) ) ).

tff(fact_208_power2__le__imp__le,axiom,
    ! [X_45: int,Y_33: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_33))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_45),Y_33)) ) ) ).

tff(fact_209_power2__le__imp__le,axiom,
    ! [X_45: nat,Y_33: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_33))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_45),Y_33)) ) ) ).

tff(fact_210_power2__le__imp__le,axiom,
    ! [X_45: real,Y_33: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_33))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_45),Y_33)) ) ) ).

tff(fact_211_power2__le__imp__le,axiom,
    ! [X_45: code_code_numeral,Y_33: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_33))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_45),Y_33)) ) ) ).

tff(fact_212_power2__le__imp__le,axiom,
    ! [X_45: quickcheck_code_int,Y_33: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_33))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_45),Y_33)) ) ) ).

tff(fact_213_power2__le__imp__le,axiom,
    ! [X_45: rat,Y_33: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_33))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_45),Y_33)) ) ) ).

tff(fact_214_zero__le__power2,axiom,
    ! [A_238: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_238),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_215_zero__le__power2,axiom,
    ! [A_238: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_238),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_216_zero__le__power2,axiom,
    ! [A_238: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_238),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_217_power2__less__imp__less,axiom,
    ! [X_44: int,Y_32: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_32))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_44),Y_32)) ) ) ).

tff(fact_218_power2__less__imp__less,axiom,
    ! [X_44: nat,Y_32: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_32))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_44),Y_32)) ) ) ).

tff(fact_219_power2__less__imp__less,axiom,
    ! [X_44: real,Y_32: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_32))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_44),Y_32)) ) ) ).

tff(fact_220_power2__less__imp__less,axiom,
    ! [X_44: code_code_numeral,Y_32: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_32))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_44),Y_32)) ) ) ).

tff(fact_221_power2__less__imp__less,axiom,
    ! [X_44: quickcheck_code_int,Y_32: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_32))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_44),Y_32)) ) ) ).

tff(fact_222_power2__less__imp__less,axiom,
    ! [X_44: rat,Y_32: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_32))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_44),Y_32)) ) ) ).

tff(fact_223_sum__power2__le__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))
    <=> ( ( X_1 = zero_zero_int )
        & ( Y_2 = zero_zero_int ) ) ) ).

tff(fact_224_sum__power2__le__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real))
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_225_sum__power2__le__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat))
    <=> ( ( X_1 = zero_zero_rat )
        & ( Y_2 = zero_zero_rat ) ) ) ).

tff(fact_226_sum__power2__ge__zero,axiom,
    ! [X_43: int,Y_31: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_43),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_31),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_227_sum__power2__ge__zero,axiom,
    ! [X_43: real,Y_31: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_43),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_31),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_228_sum__power2__ge__zero,axiom,
    ! [X_43: rat,Y_31: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_43),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_31),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_229_sum__squares__gt__zero__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))))
    <=> ( ( X_1 != zero_zero_int )
        | ( Y_2 != zero_zero_int ) ) ) ).

tff(fact_230_sum__squares__gt__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))))
    <=> ( ( X_1 != zero_zero_real )
        | ( Y_2 != zero_zero_real ) ) ) ).

tff(fact_231_sum__squares__gt__zero__iff,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))))
    <=> ( ( X_1 != zero_zero_rat )
        | ( Y_2 != zero_zero_rat ) ) ) ).

tff(fact_232_not__sum__squares__lt__zero,axiom,
    ! [X_42: int,Y_30: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_42),X_42)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_30),Y_30))),zero_zero_int)) ).

tff(fact_233_not__sum__squares__lt__zero,axiom,
    ! [X_42: real,Y_30: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_42),X_42)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_30),Y_30))),zero_zero_real)) ).

tff(fact_234_not__sum__squares__lt__zero,axiom,
    ! [X_42: rat,Y_30: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_42),X_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_30),Y_30))),zero_zero_rat)) ).

tff(fact_235_mult__numeral__1,axiom,
    ! [A_237: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))),A_237) = A_237 ).

tff(fact_236_mult__numeral__1,axiom,
    ! [A_237: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_237) = A_237 ).

tff(fact_237_mult__numeral__1,axiom,
    ! [A_237: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))),A_237) = A_237 ).

tff(fact_238_mult__numeral__1,axiom,
    ! [A_237: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))),A_237) = A_237 ).

tff(fact_239_mult__numeral__1__right,axiom,
    ! [A_236: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_236),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))) = A_236 ).

tff(fact_240_mult__numeral__1__right,axiom,
    ! [A_236: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_236),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))) = A_236 ).

tff(fact_241_mult__numeral__1__right,axiom,
    ! [A_236: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_236),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))) = A_236 ).

tff(fact_242_mult__numeral__1__right,axiom,
    ! [A_236: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_236),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))) = A_236 ).

tff(fact_243_one__is__num__one,axiom,
    one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).

tff(fact_244_mult__Bit1,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit1,K_2)),L_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))),L_2) ).

tff(fact_245_pos__zmult__eq__1__iff,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1) = one_one_int )
      <=> ( ( Ma = one_one_int )
          & ( N_1 = one_one_int ) ) ) ) ).

tff(fact_246_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

tff(fact_247_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

tff(fact_248_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

tff(fact_249_rel__simps_I46_J,axiom,
    ! [K_2: int] : hAPP_int_int(bit1,K_2) != pls ).

tff(fact_250_rel__simps_I39_J,axiom,
    ! [L_2: int] : pls != hAPP_int_int(bit1,L_2) ).

tff(fact_251_rel__simps_I50_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(bit1,K_2) != hAPP_int_int(bit0,L_2) ).

tff(fact_252_rel__simps_I49_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(bit0,K_2) != hAPP_int_int(bit1,L_2) ).

tff(fact_253_rel__simps_I44_J,axiom,
    ! [K_1: int] :
      ( ( hAPP_int_int(bit0,K_1) = pls )
    <=> ( K_1 = pls ) ) ).

tff(fact_254_rel__simps_I38_J,axiom,
    ! [L: int] :
      ( ( pls = hAPP_int_int(bit0,L) )
    <=> ( pls = L ) ) ).

tff(fact_255_Bit0__Pls,axiom,
    hAPP_int_int(bit0,pls) = pls ).

tff(fact_256_Pls__def,axiom,
    pls = zero_zero_int ).

tff(fact_257_less__int__code_I16_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_258_rel__simps_I17_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_259_rel__simps_I2_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).

tff(fact_260_less__int__code_I13_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_261_rel__simps_I14_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_262_int__0__neq__1,axiom,
    zero_zero_int != one_one_int ).

tff(fact_263_add__Pls__right,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),pls) = K_2 ).

tff(fact_264_add__Pls,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_2) = K_2 ).

tff(fact_265_add__Bit0__Bit0,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).

tff(fact_266_Bit0__def,axiom,
    ! [K_2: int] : hAPP_int_int(bit0,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),K_2) ).

tff(fact_267_zadd__0__right,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),zero_zero_int) = Z_1 ).

tff(fact_268_zadd__0,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1) = Z_1 ).

tff(fact_269_zadd__strict__right__mono,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J_2),K_2))) ) ).

tff(fact_270_double__number__of__Bit0,axiom,
    ! [W_12: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),hAPP_int_int(number_number_of_int,W_12)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_12)) ).

tff(fact_271_double__number__of__Bit0,axiom,
    ! [W_12: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),hAPP_int_real(number267125858f_real,W_12)) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_12)) ).

tff(fact_272_double__number__of__Bit0,axiom,
    ! [W_12: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),hAPP_int_complex(number528085621omplex,W_12)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_12)) ).

tff(fact_273_double__number__of__Bit0,axiom,
    ! [W_12: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),hAPP_int_rat(number_number_of_rat,W_12)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_12)) ).

tff(fact_274_power3__eq__cube,axiom,
    ! [A_235: code_code_numeral] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_235),A_235)),A_235) ).

tff(fact_275_power3__eq__cube,axiom,
    ! [A_235: quickcheck_code_int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_235),A_235)),A_235) ).

tff(fact_276_power3__eq__cube,axiom,
    ! [A_235: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_235),A_235)),A_235) ).

tff(fact_277_power3__eq__cube,axiom,
    ! [A_235: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_235),A_235)),A_235) ).

tff(fact_278_power3__eq__cube,axiom,
    ! [A_235: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_235),A_235)),A_235) ).

tff(fact_279_power3__eq__cube,axiom,
    ! [A_235: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_235),A_235)),A_235) ).

tff(fact_280_power3__eq__cube,axiom,
    ! [A_235: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_235),A_235)),A_235) ).

tff(fact_281_semiring__mult__2,axiom,
    ! [Z_14: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_14),Z_14) ).

tff(fact_282_semiring__mult__2,axiom,
    ! [Z_14: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_14),Z_14) ).

tff(fact_283_semiring__mult__2,axiom,
    ! [Z_14: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_14),Z_14) ).

tff(fact_284_semiring__mult__2,axiom,
    ! [Z_14: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_14),Z_14) ).

tff(fact_285_semiring__mult__2,axiom,
    ! [Z_14: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_14),Z_14) ).

tff(fact_286_mult__2,axiom,
    ! [Z_13: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_13),Z_13) ).

tff(fact_287_mult__2,axiom,
    ! [Z_13: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_13),Z_13) ).

tff(fact_288_mult__2,axiom,
    ! [Z_13: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_13),Z_13) ).

tff(fact_289_mult__2,axiom,
    ! [Z_13: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_13),Z_13) ).

tff(fact_290_semiring__mult__2__right,axiom,
    ! [Z_12: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_12),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_12),Z_12) ).

tff(fact_291_semiring__mult__2__right,axiom,
    ! [Z_12: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_12),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_12),Z_12) ).

tff(fact_292_semiring__mult__2__right,axiom,
    ! [Z_12: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_12),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_12),Z_12) ).

tff(fact_293_semiring__mult__2__right,axiom,
    ! [Z_12: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_12),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_12),Z_12) ).

tff(fact_294_semiring__mult__2__right,axiom,
    ! [Z_12: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_12),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_12),Z_12) ).

tff(fact_295_mult__2__right,axiom,
    ! [Z_11: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_11),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_11),Z_11) ).

tff(fact_296_mult__2__right,axiom,
    ! [Z_11: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_11),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_11),Z_11) ).

tff(fact_297_mult__2__right,axiom,
    ! [Z_11: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_11),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_11),Z_11) ).

tff(fact_298_mult__2__right,axiom,
    ! [Z_11: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_11),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_11),Z_11) ).

tff(fact_299_power2__eq__square,axiom,
    ! [A_234: code_code_numeral] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_234),A_234) ).

tff(fact_300_power2__eq__square,axiom,
    ! [A_234: quickcheck_code_int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_234),A_234) ).

tff(fact_301_power2__eq__square,axiom,
    ! [A_234: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_234),A_234) ).

tff(fact_302_power2__eq__square,axiom,
    ! [A_234: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_234),A_234) ).

tff(fact_303_power2__eq__square,axiom,
    ! [A_234: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_234),A_234) ).

tff(fact_304_power2__eq__square,axiom,
    ! [A_234: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_234),A_234) ).

tff(fact_305_power2__eq__square,axiom,
    ! [A_234: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_234),A_234) ).

tff(fact_306_even__less__0__iff,axiom,
    ! [A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ).

tff(fact_307_even__less__0__iff,axiom,
    ! [A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)) ) ).

tff(fact_308_even__less__0__iff,axiom,
    ! [A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)) ) ).

tff(fact_309_semiring__numeral__0__eq__0,axiom,
    hAPP_int_int(number_number_of_int,pls) = zero_zero_int ).

tff(fact_310_semiring__numeral__0__eq__0,axiom,
    hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).

tff(fact_311_semiring__numeral__0__eq__0,axiom,
    hAPP_int_real(number267125858f_real,pls) = zero_zero_real ).

tff(fact_312_semiring__numeral__0__eq__0,axiom,
    hAPP_int_complex(number528085621omplex,pls) = zero_zero_complex ).

tff(fact_313_semiring__numeral__0__eq__0,axiom,
    hAPP_int_rat(number_number_of_rat,pls) = zero_zero_rat ).

tff(fact_314_number__of__Pls,axiom,
    hAPP_int_int(number_number_of_int,pls) = zero_zero_int ).

tff(fact_315_number__of__Pls,axiom,
    hAPP_int_real(number267125858f_real,pls) = zero_zero_real ).

tff(fact_316_number__of__Pls,axiom,
    hAPP_int_complex(number528085621omplex,pls) = zero_zero_complex ).

tff(fact_317_number__of__Pls,axiom,
    hAPP_int_rat(number_number_of_rat,pls) = zero_zero_rat ).

tff(fact_318_semiring__norm_I112_J,axiom,
    zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).

tff(fact_319_semiring__norm_I112_J,axiom,
    zero_zero_real = hAPP_int_real(number267125858f_real,pls) ).

tff(fact_320_semiring__norm_I112_J,axiom,
    zero_zero_complex = hAPP_int_complex(number528085621omplex,pls) ).

tff(fact_321_semiring__norm_I112_J,axiom,
    zero_zero_rat = hAPP_int_rat(number_number_of_rat,pls) ).

tff(fact_322_add__numeral__0,axiom,
    ! [A_233: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,pls)),A_233) = A_233 ).

tff(fact_323_add__numeral__0,axiom,
    ! [A_233: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,pls)),A_233) = A_233 ).

tff(fact_324_add__numeral__0,axiom,
    ! [A_233: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,pls)),A_233) = A_233 ).

tff(fact_325_add__numeral__0,axiom,
    ! [A_233: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,pls)),A_233) = A_233 ).

tff(fact_326_add__numeral__0__right,axiom,
    ! [A_232: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_232),hAPP_int_int(number_number_of_int,pls)) = A_232 ).

tff(fact_327_add__numeral__0__right,axiom,
    ! [A_232: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_232),hAPP_int_real(number267125858f_real,pls)) = A_232 ).

tff(fact_328_add__numeral__0__right,axiom,
    ! [A_232: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_232),hAPP_int_complex(number528085621omplex,pls)) = A_232 ).

tff(fact_329_add__numeral__0__right,axiom,
    ! [A_232: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_232),hAPP_int_rat(number_number_of_rat,pls)) = A_232 ).

tff(fact_330_less__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_331_less__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_332_less__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_333_add__number__of__left,axiom,
    ! [V_13: int,W_11: int,Z_10: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_11)),Z_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10) ).

tff(fact_334_add__number__of__left,axiom,
    ! [V_13: int,W_11: int,Z_10: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_13)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_11)),Z_10)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10) ).

tff(fact_335_add__number__of__left,axiom,
    ! [V_13: int,W_11: int,Z_10: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_13)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_11)),Z_10)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10) ).

tff(fact_336_add__number__of__left,axiom,
    ! [V_13: int,W_11: int,Z_10: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_13)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_11)),Z_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10) ).

tff(fact_337_add__number__of__eq,axiom,
    ! [V_12: int,W_10: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_12)),hAPP_int_int(number_number_of_int,W_10)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).

tff(fact_338_add__number__of__eq,axiom,
    ! [V_12: int,W_10: int] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_12)),hAPP_int_real(number267125858f_real,W_10)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).

tff(fact_339_add__number__of__eq,axiom,
    ! [V_12: int,W_10: int] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_12)),hAPP_int_complex(number528085621omplex,W_10)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).

tff(fact_340_add__number__of__eq,axiom,
    ! [V_12: int,W_10: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_12)),hAPP_int_rat(number_number_of_rat,W_10)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) ).

tff(fact_341_number__of__add,axiom,
    ! [V_11: int,W_9: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_11)),hAPP_int_int(number_number_of_int,W_9)) ).

tff(fact_342_number__of__add,axiom,
    ! [V_11: int,W_9: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_11)),hAPP_int_real(number267125858f_real,W_9)) ).

tff(fact_343_number__of__add,axiom,
    ! [V_11: int,W_9: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_11)),hAPP_int_complex(number528085621omplex,W_9)) ).

tff(fact_344_number__of__add,axiom,
    ! [V_11: int,W_9: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_11)),hAPP_int_rat(number_number_of_rat,W_9)) ).

tff(fact_345_rel__simps_I12_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).

tff(fact_346_less__int__code_I15_J,axiom,
    ! [K1: int,K2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

tff(fact_347_rel__simps_I16_J,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) ) ).

tff(fact_348_bin__less__0__simps_I4_J,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,W_1)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).

tff(fact_349_rel__simps_I10_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) ) ).

tff(fact_350_rel__simps_I4_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_1)) ) ).

tff(fact_351_bin__less__0__simps_I1_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).

tff(fact_352_bin__less__0__simps_I3_J,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,W_1)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) ) ).

tff(fact_353_add__Bit1__Bit0,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).

tff(fact_354_add__Bit0__Bit1,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) ).

tff(fact_355_int__0__less__1,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

tff(fact_356_Bit1__def,axiom,
    ! [K_2: int] : hAPP_int_int(bit1,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_2)),K_2) ).

tff(fact_357_odd__nonzero,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)),Z_1) != zero_zero_int ).

tff(fact_358_zless__add1__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
        | ( W_1 = Z_2 ) ) ) ).

tff(fact_359_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_i769753017umeral(number1443263063umeral,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,W_8)),hAPP_i769753017umeral(number1443263063umeral,W_8)) ).

tff(fact_360_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_i1732201573de_int(number1226105091de_int,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,W_8)),hAPP_i1732201573de_int(number1226105091de_int,W_8)) ).

tff(fact_361_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_8)),hAPP_int_rat(number_number_of_rat,W_8)) ).

tff(fact_362_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_8)),hAPP_int_int(number_number_of_int,W_8)) ).

tff(fact_363_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,W_8)),hAPP_int_nat(number_number_of_nat,W_8)) ).

tff(fact_364_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_8)),hAPP_int_real(number267125858f_real,W_8)) ).

tff(fact_365_power2__eq__square__number__of,axiom,
    ! [W_8: int] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_8)),hAPP_int_complex(number528085621omplex,W_8)) ).

tff(fact_366_power2__sum,axiom,
    ! [X_41: int,Y_29: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) ).

tff(fact_367_power2__sum,axiom,
    ! [X_41: nat,Y_29: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) ).

tff(fact_368_power2__sum,axiom,
    ! [X_41: real,Y_29: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) ).

tff(fact_369_power2__sum,axiom,
    ! [X_41: complex,Y_29: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) ).

tff(fact_370_power2__sum,axiom,
    ! [X_41: rat,Y_29: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) ).

tff(fact_371_number__of__Bit0,axiom,
    ! [W_7: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_7)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),hAPP_int_int(number_number_of_int,W_7))),hAPP_int_int(number_number_of_int,W_7)) ).

tff(fact_372_number__of__Bit0,axiom,
    ! [W_7: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_7)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),hAPP_int_real(number267125858f_real,W_7))),hAPP_int_real(number267125858f_real,W_7)) ).

tff(fact_373_number__of__Bit0,axiom,
    ! [W_7: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_7)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),hAPP_int_complex(number528085621omplex,W_7))),hAPP_int_complex(number528085621omplex,W_7)) ).

tff(fact_374_number__of__Bit0,axiom,
    ! [W_7: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,W_7))),hAPP_int_rat(number_number_of_rat,W_7)) ).

tff(fact_375_number__of__Bit1,axiom,
    ! [W_6: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_6)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_6))),hAPP_int_int(number_number_of_int,W_6)) ).

tff(fact_376_number__of__Bit1,axiom,
    ! [W_6: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,W_6)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_6))),hAPP_int_real(number267125858f_real,W_6)) ).

tff(fact_377_number__of__Bit1,axiom,
    ! [W_6: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,W_6)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_6))),hAPP_int_complex(number528085621omplex,W_6)) ).

tff(fact_378_number__of__Bit1,axiom,
    ! [W_6: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,W_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_6))),hAPP_int_rat(number_number_of_rat,W_6)) ).

tff(fact_379_semiring__numeral__1__eq__1,axiom,
    hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) = one_one_int ).

tff(fact_380_semiring__numeral__1__eq__1,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).

tff(fact_381_semiring__numeral__1__eq__1,axiom,
    hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) = one_one_real ).

tff(fact_382_semiring__numeral__1__eq__1,axiom,
    hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) = one_one_complex ).

tff(fact_383_semiring__numeral__1__eq__1,axiom,
    hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) = one_one_rat ).

tff(fact_384_numeral__1__eq__1,axiom,
    hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) = one_one_int ).

tff(fact_385_numeral__1__eq__1,axiom,
    hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) = one_one_real ).

tff(fact_386_numeral__1__eq__1,axiom,
    hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) = one_one_complex ).

tff(fact_387_numeral__1__eq__1,axiom,
    hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) = one_one_rat ).

tff(fact_388_semiring__norm_I110_J,axiom,
    one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).

tff(fact_389_semiring__norm_I110_J,axiom,
    one_one_real = hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) ).

tff(fact_390_semiring__norm_I110_J,axiom,
    one_one_complex = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) ).

tff(fact_391_semiring__norm_I110_J,axiom,
    one_one_rat = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) ).

tff(fact_392_odd__less__0,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).

tff(fact_393_less__special_I3_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),pls)) ) ).

tff(fact_394_less__special_I3_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),zero_zero_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),pls)) ) ).

tff(fact_395_less__special_I3_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),zero_zero_rat))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),pls)) ) ).

tff(fact_396_less__special_I1_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).

tff(fact_397_less__special_I1_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).

tff(fact_398_less__special_I1_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) ) ).

tff(fact_399_semiring__one__add__one__is__two,axiom,
    hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_400_semiring__one__add__one__is__two,axiom,
    hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_401_semiring__one__add__one__is__two,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_402_semiring__one__add__one__is__two,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_403_semiring__one__add__one__is__two,axiom,
    hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_404_one__add__one__is__two,axiom,
    hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_405_one__add__one__is__two,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_406_one__add__one__is__two,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_407_one__add__one__is__two,axiom,
    hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_408_less__special_I4_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),one_one_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(bit1,pls))) ) ).

tff(fact_409_less__special_I4_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),one_one_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(bit1,pls))) ) ).

tff(fact_410_less__special_I4_J,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),one_one_rat))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(bit1,pls))) ) ).

tff(fact_411_less__special_I2_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).

tff(fact_412_less__special_I2_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).

tff(fact_413_less__special_I2_J,axiom,
    ! [Y_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)) ) ).

tff(fact_414__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int))) ).

tff(fact_415_zadd__power3,axiom,
    ! [A_3: int,B_2: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_416_zadd__power2,axiom,
    ! [A_3: int,B_2: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_417_int__pos__lt__two__imp__zero__or__one,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => ( ( X = zero_zero_int )
          | ( X = one_one_int ) ) ) ) ).

tff(fact_418_s0p,axiom,
    ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),s))
    & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,s),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
    & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),s),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ) ).

tff(fact_419_cube__square,axiom,
    ! [A_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_420_power2__ge__self,axiom,
    ! [X: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_421_self__quotient__aux1,axiom,
    ! [R_3: int,Q_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),A_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)) ) ) ) ).

tff(fact_422_self__quotient__aux2,axiom,
    ! [R_3: int,Q_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)) ) ) ) ).

tff(fact_423_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_424_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: code_code_numeral,N_69: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_40),N_69)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_40),N_69)) ).

tff(fact_425_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: quickcheck_code_int,N_69: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_40),N_69)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_40),N_69)) ).

tff(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: rat,N_69: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_40),N_69)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_40),N_69)) ).

tff(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: int,N_69: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_40),N_69)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_40),N_69)) ).

tff(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: nat,N_69: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_40),N_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_40),N_69)) ).

tff(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: real,N_69: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_40),N_69)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_40),N_69)) ).

tff(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_40: complex,N_69: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_40),N_69)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_40),N_69)) ).

tff(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_39),X_39) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_39),X_39) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_39),X_39) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_39),X_39) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_39),X_39) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_39),X_39) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_39: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_39),X_39) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_438__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_439__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
    ? [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
      & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
      & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),X_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
      & ! [Y_1: int] :
          ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),Y_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_440__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
    ~ ! [S_2: int] :
        ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S_2))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ) ).

tff(fact_441_s1,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_442_zero__less__power__nat__eq,axiom,
    ! [X_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))
    <=> ( ( N_1 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) ) ) ).

tff(fact_443_zprime__zdvd__power,axiom,
    ! [A_3: int,N: nat,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3)) ) ) ).

tff(fact_444_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: quickcheck_code_int,P_8: nat,Q_12: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_38),P_8)),Q_12) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: code_code_numeral,P_8: nat,Q_12: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_38),P_8)),Q_12) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: rat,P_8: nat,Q_12: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_38),P_8)),Q_12) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: int,P_8: nat,Q_12: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_38),P_8)),Q_12) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_448_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: nat,P_8: nat,Q_12: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_38),P_8)),Q_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_449_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: real,P_8: nat,Q_12: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_38),P_8)),Q_12) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_450_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_38: complex,P_8: nat,Q_12: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_38),P_8)),Q_12) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) ).

tff(fact_451_zprime__power__zdvd__cancel__right,axiom,
    ! [N: nat,A_3: int,B_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),B_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),A_3)) ) ) ) ).

tff(fact_452_zprime__power__zdvd__cancel__left,axiom,
    ! [N: nat,B_2: int,A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),B_2)) ) ) ) ).

tff(fact_453_zpower__zpower,axiom,
    ! [X: int,Y: nat,Z_1: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),Z_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1)) ).

tff(fact_454_zdvd__not__zless,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) ) ) ).

tff(fact_455_zdvd__antisym__nonneg,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
           => ( M = N ) ) ) ) ) ).

tff(fact_456_zdvd__mult__cancel,axiom,
    ! [K_2: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)))
     => ( ( K_2 != zero_zero_int )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)) ) ) ).

tff(fact_457_zdvd__reduce,axiom,
    ! [K_1: int,N_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma))))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) ) ).

tff(fact_458_zdvd__period,axiom,
    ! [C: int,X_1: int,Ta: int,A: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),D))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D))),Ta))) ) ) ).

tff(fact_459_zprime__2,axiom,
    hBOOL(hAPP_int_bool(zprime,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_460_zdvd__imp__le,axiom,
    ! [Z_1: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N)) ) ) ).

tff(fact_461_is__mult__sum2sq,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(twoSqu1425020248sum2sq,X))
     => ( hBOOL(hAPP_int_bool(twoSqu1425020248sum2sq,Y))
       => hBOOL(hAPP_int_bool(twoSqu1425020248sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_462_le__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)))
    <=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),V_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) ) ) ).

tff(fact_463_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: code_code_numeral,Ly_4: code_code_numeral,Rx_6: code_code_numeral,Ry_4: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Ly_4)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_6),Ry_4)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Rx_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_4),Ry_4)) ).

tff(fact_464_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ly_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Rx_6)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Ry_4)) ).

tff(fact_465_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ly_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Rx_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Ry_4)) ).

tff(fact_466_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: real,Ly_4: real,Rx_6: real,Ry_4: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ly_4)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Rx_6)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Ry_4)) ).

tff(fact_467_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: complex,Ly_4: complex,Rx_6: complex,Ry_4: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ly_4)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Rx_6)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Ry_4)) ).

tff(fact_468_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: quickcheck_code_int,Ly_4: quickcheck_code_int,Rx_6: quickcheck_code_int,Ry_4: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ly_4)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Rx_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Ry_4)) ).

tff(fact_469_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [Lx_6: rat,Ly_4: rat,Rx_6: rat,Ry_4: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ly_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),Ry_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Rx_6)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_4),Ry_4)) ).

tff(fact_470_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: code_code_numeral,Ly_3: code_code_numeral,Rx_5: code_code_numeral,Ry_3: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),Ry_3)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_471_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_472_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_473_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: real,Ly_3: real,Rx_5: real,Ry_3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_474_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: complex,Ly_3: complex,Rx_5: complex,Ry_3: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_475_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: quickcheck_code_int,Ly_3: quickcheck_code_int,Rx_5: quickcheck_code_int,Ry_3: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_476_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [Lx_5: rat,Ly_3: rat,Rx_5: rat,Ry_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),Ry_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),Ry_3)) ).

tff(fact_477_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: code_code_numeral,Ly_2: code_code_numeral,Rx_4: code_code_numeral,Ry_2: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),Ly_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2))) ).

tff(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2))) ).

tff(fact_479_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2))) ).

tff(fact_480_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: real,Ly_2: real,Rx_4: real,Ry_2: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2))) ).

tff(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: complex,Ly_2: complex,Rx_4: complex,Ry_2: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2))) ).

tff(fact_482_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: quickcheck_code_int,Ly_2: quickcheck_code_int,Rx_4: quickcheck_code_int,Ry_2: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2))) ).

tff(fact_483_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [Lx_4: rat,Ly_2: rat,Rx_4: rat,Ry_2: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2))) ).

tff(fact_484_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: code_code_numeral,Ly_1: code_code_numeral,Rx_3: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Ly_1)),Rx_3) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Rx_3)),Ly_1) ).

tff(fact_485_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: int,Ly_1: int,Rx_3: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_1)),Rx_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_1) ).

tff(fact_486_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: nat,Ly_1: nat,Rx_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_1) ).

tff(fact_487_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: real,Ly_1: real,Rx_3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_1)),Rx_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Rx_3)),Ly_1) ).

tff(fact_488_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: complex,Ly_1: complex,Rx_3: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_1)),Rx_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_1) ).

tff(fact_489_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: quickcheck_code_int,Ly_1: quickcheck_code_int,Rx_3: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_1)),Rx_3) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Rx_3)),Ly_1) ).

tff(fact_490_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [Lx_3: rat,Ly_1: rat,Rx_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Ly_1)),Rx_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Rx_3)),Ly_1) ).

tff(fact_491_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: code_code_numeral,Ly: code_code_numeral,Rx_2: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),Ly)),Rx_2) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly),Rx_2)) ).

tff(fact_492_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: int,Ly: int,Rx_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly)),Rx_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Rx_2)) ).

tff(fact_493_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: nat,Ly: nat,Rx_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly)),Rx_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Rx_2)) ).

tff(fact_494_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: real,Ly: real,Rx_2: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly)),Rx_2) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Rx_2)) ).

tff(fact_495_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: complex,Ly: complex,Rx_2: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly)),Rx_2) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Rx_2)) ).

tff(fact_496_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: quickcheck_code_int,Ly: quickcheck_code_int,Rx_2: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly)),Rx_2) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Rx_2)) ).

tff(fact_497_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [Lx_2: rat,Ly: rat,Rx_2: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),Ly)),Rx_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Rx_2)) ).

tff(fact_498_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: code_code_numeral,Rx_1: code_code_numeral,Ry_1: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_1),Ry_1)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),Rx_1)),Ry_1) ).

tff(fact_499_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: int,Rx_1: int,Ry_1: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Rx_1)),Ry_1) ).

tff(fact_500_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: nat,Rx_1: nat,Ry_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Rx_1)),Ry_1) ).

tff(fact_501_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: real,Rx_1: real,Ry_1: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Rx_1)),Ry_1) ).

tff(fact_502_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: complex,Rx_1: complex,Ry_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Rx_1)),Ry_1) ).

tff(fact_503_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: quickcheck_code_int,Rx_1: quickcheck_code_int,Ry_1: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Rx_1)),Ry_1) ).

tff(fact_504_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [Lx_1: rat,Rx_1: rat,Ry_1: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),Ry_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Rx_1)),Ry_1) ).

tff(fact_505_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: code_code_numeral,Rx: code_code_numeral,Ry: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),Ry)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),Ry)) ).

tff(fact_506_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: int,Rx: int,Ry: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ry)) ).

tff(fact_507_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: nat,Rx: nat,Ry: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ry)) ).

tff(fact_508_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: real,Rx: real,Ry: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ry)) ).

tff(fact_509_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: complex,Rx: complex,Ry: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ry)) ).

tff(fact_510_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: quickcheck_code_int,Rx: quickcheck_code_int,Ry: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ry)) ).

tff(fact_511_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [Lx: rat,Rx: rat,Ry: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ry)) ).

tff(fact_512_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: code_code_numeral,B_165: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_231),B_165) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_165),A_231) ).

tff(fact_513_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: int,B_165: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_231),B_165) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_165),A_231) ).

tff(fact_514_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: nat,B_165: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_231),B_165) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_165),A_231) ).

tff(fact_515_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: real,B_165: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_231),B_165) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_165),A_231) ).

tff(fact_516_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: complex,B_165: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_231),B_165) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_165),A_231) ).

tff(fact_517_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: quickcheck_code_int,B_165: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_231),B_165) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_165),A_231) ).

tff(fact_518_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [A_231: rat,B_165: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_231),B_165) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_165),A_231) ).

tff(fact_519_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: code_code_numeral,C_96: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_230),C_96) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_96),A_230) ).

tff(fact_520_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: int,C_96: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_230),C_96) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_96),A_230) ).

tff(fact_521_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: nat,C_96: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_230),C_96) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_96),A_230) ).

tff(fact_522_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: real,C_96: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_230),C_96) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_96),A_230) ).

tff(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: complex,C_96: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_230),C_96) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_96),A_230) ).

tff(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: quickcheck_code_int,C_96: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_230),C_96) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_96),A_230) ).

tff(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [A_230: rat,C_96: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_230),C_96) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_96),A_230) ).

tff(fact_526_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: code_code_numeral,C_95: code_code_numeral,D_20: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_229),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_95),D_20)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_95),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_229),D_20)) ).

tff(fact_527_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: int,C_95: int,D_20: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_229),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_95),D_20)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_95),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_229),D_20)) ).

tff(fact_528_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: nat,C_95: nat,D_20: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_229),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_95),D_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_95),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_229),D_20)) ).

tff(fact_529_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: real,C_95: real,D_20: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_229),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_95),D_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_95),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_229),D_20)) ).

tff(fact_530_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: complex,C_95: complex,D_20: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_229),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_95),D_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_95),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_229),D_20)) ).

tff(fact_531_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: quickcheck_code_int,C_95: quickcheck_code_int,D_20: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_229),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_95),D_20)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_95),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_229),D_20)) ).

tff(fact_532_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [A_229: rat,C_95: rat,D_20: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_229),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_95),D_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_95),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_229),D_20)) ).

tff(fact_533_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: code_code_numeral,C_94: code_code_numeral,D_19: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_228),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_94),D_19)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_228),C_94)),D_19) ).

tff(fact_534_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: int,C_94: int,D_19: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_228),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_94),D_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_228),C_94)),D_19) ).

tff(fact_535_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: nat,C_94: nat,D_19: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_228),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_94),D_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_228),C_94)),D_19) ).

tff(fact_536_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: real,C_94: real,D_19: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_228),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_94),D_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_228),C_94)),D_19) ).

tff(fact_537_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: complex,C_94: complex,D_19: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_228),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_94),D_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_228),C_94)),D_19) ).

tff(fact_538_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: quickcheck_code_int,C_94: quickcheck_code_int,D_19: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_228),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_94),D_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_228),C_94)),D_19) ).

tff(fact_539_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [A_228: rat,C_94: rat,D_19: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_228),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_94),D_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_228),C_94)),D_19) ).

tff(fact_540_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: code_code_numeral,B_164: code_code_numeral,C_93: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_227),B_164)),C_93) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_227),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_164),C_93)) ).

tff(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: int,B_164: int,C_93: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_227),B_164)),C_93) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_227),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_164),C_93)) ).

tff(fact_542_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: nat,B_164: nat,C_93: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_227),B_164)),C_93) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_227),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_164),C_93)) ).

tff(fact_543_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: real,B_164: real,C_93: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_227),B_164)),C_93) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_227),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_164),C_93)) ).

tff(fact_544_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: complex,B_164: complex,C_93: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_227),B_164)),C_93) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_227),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_164),C_93)) ).

tff(fact_545_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: quickcheck_code_int,B_164: quickcheck_code_int,C_93: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_227),B_164)),C_93) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_227),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_164),C_93)) ).

tff(fact_546_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [A_227: rat,B_164: rat,C_93: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_227),B_164)),C_93) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_227),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_164),C_93)) ).

tff(fact_547_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: code_code_numeral,B_163: code_code_numeral,C_92: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_226),B_163)),C_92) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_226),C_92)),B_163) ).

tff(fact_548_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: int,B_163: int,C_92: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),B_163)),C_92) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),C_92)),B_163) ).

tff(fact_549_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: nat,B_163: nat,C_92: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),B_163)),C_92) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),C_92)),B_163) ).

tff(fact_550_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: real,B_163: real,C_92: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),B_163)),C_92) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),C_92)),B_163) ).

tff(fact_551_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: complex,B_163: complex,C_92: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),B_163)),C_92) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),C_92)),B_163) ).

tff(fact_552_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: quickcheck_code_int,B_163: quickcheck_code_int,C_92: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),B_163)),C_92) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),C_92)),B_163) ).

tff(fact_553_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [A_226: rat,B_163: rat,C_92: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),B_163)),C_92) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),C_92)),B_163) ).

tff(fact_554_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: code_code_numeral,B_162: code_code_numeral,C_91: code_code_numeral,D_18: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_225),B_162)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_91),D_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_225),C_91)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_162),D_18)) ).

tff(fact_555_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: int,B_162: int,C_91: int,D_18: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_225),B_162)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_91),D_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_225),C_91)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_162),D_18)) ).

tff(fact_556_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: nat,B_162: nat,C_91: nat,D_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_225),B_162)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_91),D_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_225),C_91)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_162),D_18)) ).

tff(fact_557_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: real,B_162: real,C_91: real,D_18: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_225),B_162)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_91),D_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_225),C_91)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_162),D_18)) ).

tff(fact_558_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: complex,B_162: complex,C_91: complex,D_18: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_225),B_162)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_91),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_225),C_91)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_162),D_18)) ).

tff(fact_559_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: quickcheck_code_int,B_162: quickcheck_code_int,C_91: quickcheck_code_int,D_18: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_225),B_162)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_91),D_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_225),C_91)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_162),D_18)) ).

tff(fact_560_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [A_225: rat,B_162: rat,C_91: rat,D_18: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_225),B_162)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_91),D_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_225),C_91)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_162),D_18)) ).

tff(fact_561_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: quickcheck_code_int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_37),one_one_nat) = X_37 ).

tff(fact_562_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: code_code_numeral] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_37),one_one_nat) = X_37 ).

tff(fact_563_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_37),one_one_nat) = X_37 ).

tff(fact_564_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_37),one_one_nat) = X_37 ).

tff(fact_565_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_37),one_one_nat) = X_37 ).

tff(fact_566_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_37),one_one_nat) = X_37 ).

tff(fact_567_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_37: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_37),one_one_nat) = X_37 ).

tff(fact_568_zero__less__power__nat__eq__number__of,axiom,
    ! [X_1: nat,W_1: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,W_1))))
    <=> ( ( hAPP_int_nat(number_number_of_nat,W_1) = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) ) ) ).

tff(fact_569_nat__mult__2__right,axiom,
    ! [Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) ).

tff(fact_570_nat__mult__2,axiom,
    ! [Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) ).

tff(fact_571_mult__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3)) ) ) ) ).

tff(fact_572_nat__number__of__mult__left,axiom,
    ! [V_3: int,K_2: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3))),K_2) ) ) ) ).

tff(fact_573_power__even__eq,axiom,
    ! [A_224: quickcheck_code_int,N_68: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_574_power__even__eq,axiom,
    ! [A_224: code_code_numeral,N_68: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_575_power__even__eq,axiom,
    ! [A_224: rat,N_68: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_576_power__even__eq,axiom,
    ! [A_224: int,N_68: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_577_power__even__eq,axiom,
    ! [A_224: nat,N_68: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_578_power__even__eq,axiom,
    ! [A_224: real,N_68: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_579_power__even__eq,axiom,
    ! [A_224: complex,N_68: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_580_even__power__le__0__imp__0,axiom,
    ! [A_223: int,K_8: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_int))
     => ( A_223 = zero_zero_int ) ) ).

tff(fact_581_even__power__le__0__imp__0,axiom,
    ! [A_223: real,K_8: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_real))
     => ( A_223 = zero_zero_real ) ) ).

tff(fact_582_even__power__le__0__imp__0,axiom,
    ! [A_223: rat,K_8: nat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_rat))
     => ( A_223 = zero_zero_rat ) ) ).

tff(fact_583_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_222) = zero_zero_int ).

tff(fact_584_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_222) = zero_zero_nat ).

tff(fact_585_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_222) = zero_zero_real ).

tff(fact_586_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_222) = zero_z126310315umeral ).

tff(fact_587_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_222) = zero_zero_complex ).

tff(fact_588_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_222) = zero_z891286103de_int ).

tff(fact_589_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [A_222: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_222) = zero_zero_rat ).

tff(fact_590_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_221),zero_zero_int) = zero_zero_int ).

tff(fact_591_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_221),zero_zero_nat) = zero_zero_nat ).

tff(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_221),zero_zero_real) = zero_zero_real ).

tff(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_221),zero_z126310315umeral) = zero_z126310315umeral ).

tff(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_221),zero_zero_complex) = zero_zero_complex ).

tff(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_221),zero_z891286103de_int) = zero_z891286103de_int ).

tff(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [A_221: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_221),zero_zero_rat) = zero_zero_rat ).

tff(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_220) = A_220 ).

tff(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_220) = A_220 ).

tff(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_220) = A_220 ).

tff(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_220) = A_220 ).

tff(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_220) = A_220 ).

tff(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_220) = A_220 ).

tff(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [A_220: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_220) = A_220 ).

tff(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_219),zero_z126310315umeral) = A_219 ).

tff(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_219),zero_zero_int) = A_219 ).

tff(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_219),zero_zero_nat) = A_219 ).

tff(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_219),zero_zero_real) = A_219 ).

tff(fact_608_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_219),zero_zero_complex) = A_219 ).

tff(fact_609_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_219),zero_z891286103de_int) = A_219 ).

tff(fact_610_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [A_219: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_219),zero_zero_rat) = A_219 ).

tff(fact_611_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

tff(fact_612_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

tff(fact_613_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

tff(fact_614_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) )
    <=> ( A = zero_zero_complex ) ) ).

tff(fact_615_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

tff(fact_616_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: code_code_numeral,Y_28: code_code_numeral,Z_9: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_36),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Y_28),Z_9)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_36),Y_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_36),Z_9)) ).

tff(fact_617_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: int,Y_28: int,Z_9: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_36),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_28),Z_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_36),Y_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_36),Z_9)) ).

tff(fact_618_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: nat,Y_28: nat,Z_9: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_36),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_28),Z_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_36),Y_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_36),Z_9)) ).

tff(fact_619_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: real,Y_28: real,Z_9: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_28),Z_9)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),Y_28)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),Z_9)) ).

tff(fact_620_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: complex,Y_28: complex,Z_9: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_28),Z_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),Y_28)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),Z_9)) ).

tff(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: quickcheck_code_int,Y_28: quickcheck_code_int,Z_9: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_36),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_28),Z_9)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_36),Y_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_36),Z_9)) ).

tff(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_36: rat,Y_28: rat,Z_9: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_36),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_28),Z_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_36),Y_28)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_36),Z_9)) ).

tff(fact_623_crossproduct__noteq,axiom,
    ! [C: int,D: int,A: int,B: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)) ) ) ).

tff(fact_624_crossproduct__noteq,axiom,
    ! [C: nat,D: nat,A: nat,B: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

tff(fact_625_crossproduct__noteq,axiom,
    ! [C: real,D: real,A: real,B: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),D)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),D)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)) ) ) ).

tff(fact_626_crossproduct__noteq,axiom,
    ! [C: complex,D: complex,A: complex,B: complex] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),D)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),D)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C)) ) ) ).

tff(fact_627_crossproduct__noteq,axiom,
    ! [C: rat,D: rat,A: rat,B: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),D)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),D)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

tff(fact_628_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: code_code_numeral,B_161: code_code_numeral,C_90: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_218),B_161)),C_90) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_218),C_90)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_161),C_90)) ).

tff(fact_629_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: int,B_161: int,C_90: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_218),B_161)),C_90) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_218),C_90)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_161),C_90)) ).

tff(fact_630_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: nat,B_161: nat,C_90: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_218),B_161)),C_90) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_218),C_90)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_161),C_90)) ).

tff(fact_631_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: real,B_161: real,C_90: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_218),B_161)),C_90) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_218),C_90)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_161),C_90)) ).

tff(fact_632_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: complex,B_161: complex,C_90: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_218),B_161)),C_90) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_218),C_90)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_161),C_90)) ).

tff(fact_633_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: quickcheck_code_int,B_161: quickcheck_code_int,C_90: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_218),B_161)),C_90) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_218),C_90)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_161),C_90)) ).

tff(fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [A_218: rat,B_161: rat,C_90: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_218),B_161)),C_90) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_218),C_90)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_161),C_90)) ).

tff(fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: code_code_numeral,M_29: code_code_numeral,B_160: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_217),M_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_160),M_29)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_217),B_160)),M_29) ).

tff(fact_636_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: int,M_29: int,B_160: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_217),M_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_160),M_29)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_217),B_160)),M_29) ).

tff(fact_637_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: nat,M_29: nat,B_160: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_217),M_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_160),M_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_217),B_160)),M_29) ).

tff(fact_638_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: real,M_29: real,B_160: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_217),M_29)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_160),M_29)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_217),B_160)),M_29) ).

tff(fact_639_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: complex,M_29: complex,B_160: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_217),M_29)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_160),M_29)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_217),B_160)),M_29) ).

tff(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: quickcheck_code_int,M_29: quickcheck_code_int,B_160: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_217),M_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_160),M_29)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_217),B_160)),M_29) ).

tff(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [A_217: rat,M_29: rat,B_160: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_217),M_29)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_160),M_29)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_217),B_160)),M_29) ).

tff(fact_642_crossproduct__eq,axiom,
    ! [W_1: int,Y_2: int,X_1: int,Z_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Y_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Z_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_643_crossproduct__eq,axiom,
    ! [W_1: nat,Y_2: nat,X_1: nat,Z_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Z_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_644_crossproduct__eq,axiom,
    ! [W_1: real,Y_2: real,X_1: real,Z_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Y_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_645_crossproduct__eq,axiom,
    ! [W_1: complex,Y_2: complex,X_1: complex,Z_2: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Y_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_646_crossproduct__eq,axiom,
    ! [W_1: rat,Y_2: rat,X_1: rat,Z_2: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Y_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Y_2)) )
    <=> ( ( W_1 = X_1 )
        | ( Y_2 = Z_2 ) ) ) ).

tff(fact_647_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_216) = A_216 ).

tff(fact_648_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_216) = A_216 ).

tff(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_216) = A_216 ).

tff(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_216) = A_216 ).

tff(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_216) = A_216 ).

tff(fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_216) = A_216 ).

tff(fact_653_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [A_216: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_216) = A_216 ).

tff(fact_654_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_215),one_one_int) = A_215 ).

tff(fact_655_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_215),one_one_nat) = A_215 ).

tff(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_215),one_one_real) = A_215 ).

tff(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_215),one_on1645066479umeral) = A_215 ).

tff(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_215),one_one_complex) = A_215 ).

tff(fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_215),one_on1684967323de_int) = A_215 ).

tff(fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [A_215: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_215),one_one_rat) = A_215 ).

tff(fact_661_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: code_code_numeral,Y_27: code_code_numeral,Q_11: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_35),Y_27)),Q_11) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_35),Q_11)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_27),Q_11)) ).

tff(fact_662_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: quickcheck_code_int,Y_27: quickcheck_code_int,Q_11: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_35),Y_27)),Q_11) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_35),Q_11)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_27),Q_11)) ).

tff(fact_663_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: rat,Y_27: rat,Q_11: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_35),Y_27)),Q_11) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_35),Q_11)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_27),Q_11)) ).

tff(fact_664_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: int,Y_27: int,Q_11: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_35),Y_27)),Q_11) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_35),Q_11)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_27),Q_11)) ).

tff(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: nat,Y_27: nat,Q_11: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_35),Y_27)),Q_11) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_35),Q_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_27),Q_11)) ).

tff(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: real,Y_27: real,Q_11: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_35),Y_27)),Q_11) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_35),Q_11)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_27),Q_11)) ).

tff(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_35: complex,Y_27: complex,Q_11: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_35),Y_27)),Q_11) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_35),Q_11)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_27),Q_11)) ).

tff(fact_668_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: code_code_numeral,P_7: nat,Q_10: nat] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_34),P_7)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_34),Q_10)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: quickcheck_code_int,P_7: nat,Q_10: nat] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_34),P_7)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_34),Q_10)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: rat,P_7: nat,Q_10: nat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),P_7)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),Q_10)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: int,P_7: nat,Q_10: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),P_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),Q_10)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: nat,P_7: nat,Q_10: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_34),P_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_34),Q_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: real,P_7: nat,Q_10: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),P_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),Q_10)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_34: complex,P_7: nat,Q_10: nat] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_34),P_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_34),Q_10)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) ).

tff(fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_33),zero_zero_nat) = one_one_int ).

tff(fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_33),zero_zero_nat) = one_one_nat ).

tff(fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_33),zero_zero_nat) = one_one_real ).

tff(fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: code_code_numeral] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_33),zero_zero_nat) = one_on1645066479umeral ).

tff(fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_33),zero_zero_nat) = one_one_complex ).

tff(fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: quickcheck_code_int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_33),zero_zero_nat) = one_on1684967323de_int ).

tff(fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_33: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_33),zero_zero_nat) = one_one_rat ).

tff(fact_682_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

tff(fact_683_zero__le__even__power_H,axiom,
    ! [A_214: int,N_67: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_214),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_67)))) ).

tff(fact_684_zero__le__even__power_H,axiom,
    ! [A_214: real,N_67: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_214),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_67)))) ).

tff(fact_685_zero__le__even__power_H,axiom,
    ! [A_214: rat,N_67: nat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_214),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_67)))) ).

tff(fact_686_add__scale__eq__noteq,axiom,
    ! [C_89: int,D_17: int,A_213: int,B_159: int,R_8: int] :
      ( ( R_8 != zero_zero_int )
     => ( ( ( A_213 = B_159 )
          & ( C_89 != D_17 ) )
       => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_213),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),C_89)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_159),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),D_17)) ) ) ) ).

tff(fact_687_add__scale__eq__noteq,axiom,
    ! [C_89: nat,D_17: nat,A_213: nat,B_159: nat,R_8: nat] :
      ( ( R_8 != zero_zero_nat )
     => ( ( ( A_213 = B_159 )
          & ( C_89 != D_17 ) )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_213),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),C_89)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_159),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),D_17)) ) ) ) ).

tff(fact_688_add__scale__eq__noteq,axiom,
    ! [C_89: real,D_17: real,A_213: real,B_159: real,R_8: real] :
      ( ( R_8 != zero_zero_real )
     => ( ( ( A_213 = B_159 )
          & ( C_89 != D_17 ) )
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_213),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),C_89)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_159),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),D_17)) ) ) ) ).

tff(fact_689_add__scale__eq__noteq,axiom,
    ! [C_89: complex,D_17: complex,A_213: complex,B_159: complex,R_8: complex] :
      ( ( R_8 != zero_zero_complex )
     => ( ( ( A_213 = B_159 )
          & ( C_89 != D_17 ) )
       => ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_213),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),C_89)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_159),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),D_17)) ) ) ) ).

tff(fact_690_add__scale__eq__noteq,axiom,
    ! [C_89: rat,D_17: rat,A_213: rat,B_159: rat,R_8: rat] :
      ( ( R_8 != zero_zero_rat )
     => ( ( ( A_213 = B_159 )
          & ( C_89 != D_17 ) )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_213),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),C_89)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_159),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),D_17)) ) ) ) ).

tff(fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_28),M_28) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_28) ).

tff(fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_28),M_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_28) ).

tff(fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_28),M_28) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),M_28) ).

tff(fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_28),M_28) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_28) ).

tff(fact_695_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_28),M_28) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_28) ).

tff(fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_28),M_28) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int)),M_28) ).

tff(fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [M_28: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_28),M_28) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),M_28) ).

tff(fact_698_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: int,A_212: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_27),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_212),M_27)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_212),one_one_int)),M_27) ).

tff(fact_699_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: nat,A_212: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_212),M_27)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_212),one_one_nat)),M_27) ).

tff(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: real,A_212: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_27),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_212),M_27)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_212),one_one_real)),M_27) ).

tff(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: code_code_numeral,A_212: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_27),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_212),M_27)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_212),one_on1645066479umeral)),M_27) ).

tff(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: complex,A_212: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_27),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_212),M_27)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_212),one_one_complex)),M_27) ).

tff(fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: quickcheck_code_int,A_212: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_27),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_212),M_27)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_212),one_on1684967323de_int)),M_27) ).

tff(fact_704_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [M_27: rat,A_212: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_27),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_212),M_27)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_212),one_one_rat)),M_27) ).

tff(fact_705_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: int,M_26: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_211),M_26)),M_26) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_211),one_one_int)),M_26) ).

tff(fact_706_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: nat,M_26: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_211),M_26)),M_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_211),one_one_nat)),M_26) ).

tff(fact_707_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: real,M_26: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_211),M_26)),M_26) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_211),one_one_real)),M_26) ).

tff(fact_708_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: code_code_numeral,M_26: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_211),M_26)),M_26) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_211),one_on1645066479umeral)),M_26) ).

tff(fact_709_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: complex,M_26: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_211),M_26)),M_26) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_211),one_one_complex)),M_26) ).

tff(fact_710_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: quickcheck_code_int,M_26: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_211),M_26)),M_26) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_211),one_on1684967323de_int)),M_26) ).

tff(fact_711_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [A_211: rat,M_26: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_211),M_26)),M_26) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_211),one_one_rat)),M_26) ).

tff(fact_712_power__eq__0__iff__number__of,axiom,
    ! [A: int,W_1: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_713_power__eq__0__iff__number__of,axiom,
    ! [A: nat,W_1: int] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_714_power__eq__0__iff__number__of,axiom,
    ! [A: real,W_1: int] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_715_power__eq__0__iff__number__of,axiom,
    ! [A: code_code_numeral,W_1: int] :
      ( ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_z126310315umeral )
    <=> ( ( A = zero_z126310315umeral )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_716_power__eq__0__iff__number__of,axiom,
    ! [A: complex,W_1: int] :
      ( ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_717_power__eq__0__iff__number__of,axiom,
    ! [A: quickcheck_code_int,W_1: int] :
      ( ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_z891286103de_int )
    <=> ( ( A = zero_z891286103de_int )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_718_power__eq__0__iff__number__of,axiom,
    ! [A: rat,W_1: int] :
      ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & ( hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat ) ) ) ).

tff(fact_719_pos__zmult__pos,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) ) ) ).

tff(fact_720_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

tff(fact_721_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_722_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_723_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
    ! [N: nat,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).

tff(fact_724_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_int ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_int ) ) ) ).

tff(fact_725_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_nat ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_nat ) ) ) ).

tff(fact_726_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_real ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_real ) ) ) ).

tff(fact_727_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = one_on1645066479umeral ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z126310315umeral ) ) ) ).

tff(fact_728_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_complex ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_complex ) ) ) ).

tff(fact_729_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_on1684967323de_int ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z891286103de_int ) ) ) ).

tff(fact_730_power__0__left__number__of,axiom,
    ! [W_5: int] :
      ( ( ( hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat )
       => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_rat ) )
      & ( ( hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat )
       => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_rat ) ) ) ).

tff(fact_731_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_732_q__pos__lemma,axiom,
    ! [B_5: int,Q_8: int,R_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_8)) ) ) ) ).

tff(fact_733_q__neg__lemma,axiom,
    ! [B_5: int,Q_8: int,R_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),zero_zero_int)) ) ) ) ).

tff(fact_734_unique__quotient__lemma,axiom,
    ! [B_2: int,Q_8: int,R_5: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2)) ) ) ) ) ).

tff(fact_735_zdiv__mono2__lemma,axiom,
    ! [B_2: int,Q_2: int,R_3: int,B_5: int,Q_8: int,R_5: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8)) ) ) ) ) ) ) ).

tff(fact_736_unique__quotient__lemma__neg,axiom,
    ! [B_2: int,Q_8: int,R_5: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_5))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8)) ) ) ) ) ).

tff(fact_737_zdiv__mono2__neg__lemma,axiom,
    ! [B_2: int,Q_2: int,R_3: int,B_5: int,Q_8: int,R_5: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2)) ) ) ) ) ) ) ).

tff(fact_738_quartic__square__square,axiom,
    ! [X: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_739_s,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_740_Euler_Oaux____1,axiom,
    ! [Y: int,X: int,P_3: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),P_3))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ).

tff(fact_741_zprime__def,axiom,
    ! [P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_5))
        & ! [M_1: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_5)) )
           => ( ( M_1 = one_one_int )
              | ( M_1 = P_5 ) ) ) ) ) ).

tff(fact_742_prime__g__5,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ( P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) )
       => ( ( P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) ) ) ) ).

tff(fact_743_pos2,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_744_power__strict__mono,axiom,
    ! [N_66: nat,A_210: int,B_158: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_210),B_158))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_210))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_210),N_66)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_158),N_66))) ) ) ) ).

tff(fact_745_power__strict__mono,axiom,
    ! [N_66: nat,A_210: nat,B_158: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_210),B_158))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_210))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_210),N_66)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_158),N_66))) ) ) ) ).

tff(fact_746_power__strict__mono,axiom,
    ! [N_66: nat,A_210: real,B_158: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_210),B_158))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_210))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_210),N_66)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_158),N_66))) ) ) ) ).

tff(fact_747_power__strict__mono,axiom,
    ! [N_66: nat,A_210: code_code_numeral,B_158: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_210),B_158))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_210))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66))
         => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_210),N_66)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_158),N_66))) ) ) ) ).

tff(fact_748_power__strict__mono,axiom,
    ! [N_66: nat,A_210: quickcheck_code_int,B_158: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_210),B_158))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_210))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66))
         => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_210),N_66)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_158),N_66))) ) ) ) ).

tff(fact_749_power__strict__mono,axiom,
    ! [N_66: nat,A_210: rat,B_158: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_210),B_158))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_210))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_210),N_66)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_158),N_66))) ) ) ) ).

tff(fact_750_convex__bound__lt,axiom,
    ! [V_10: int,U_3: int,Y_26: int,X_32: int,A_209: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),A_209))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_26),A_209))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_10))
           => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_10) = one_one_int )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_3),X_32)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_10),Y_26))),A_209)) ) ) ) ) ) ).

tff(fact_751_convex__bound__lt,axiom,
    ! [V_10: real,U_3: real,Y_26: real,X_32: real,A_209: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_32),A_209))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_26),A_209))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_10))
           => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),V_10) = one_one_real )
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_3),X_32)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_10),Y_26))),A_209)) ) ) ) ) ) ).

tff(fact_752_convex__bound__lt,axiom,
    ! [V_10: rat,U_3: rat,Y_26: rat,X_32: rat,A_209: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_32),A_209))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_26),A_209))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_3))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_10))
           => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),V_10) = one_one_rat )
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_3),X_32)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_10),Y_26))),A_209)) ) ) ) ) ) ).

tff(fact_753_zpower__zdvd__prop2,axiom,
    ! [Y: int,N: nat,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ) ).

tff(fact_754_zprime__zdvd__zmult,axiom,
    ! [N: int,P_3: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N)) ) ) ) ) ).

tff(fact_755_zcong__zprime__prod__zero__contra,axiom,
    ! [B_2: int,A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
       => ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
            & ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) )
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3)) ) ) ) ).

tff(fact_756_zcong__zprime__prod__zero,axiom,
    ! [B_2: int,A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) ) ) ) ) ).

tff(fact_757_dvd__0__right,axiom,
    ! [A_208: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_208),zero_zero_real)) ).

tff(fact_758_dvd__0__right,axiom,
    ! [A_208: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_208),zero_z126310315umeral)) ).

tff(fact_759_dvd__0__right,axiom,
    ! [A_208: complex] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_208),zero_zero_complex)) ).

tff(fact_760_dvd__0__right,axiom,
    ! [A_208: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_208),zero_z891286103de_int)) ).

tff(fact_761_dvd__0__right,axiom,
    ! [A_208: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_208),zero_zero_rat)) ).

tff(fact_762_dvd__0__right,axiom,
    ! [A_208: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_208),zero_zero_int)) ).

tff(fact_763_dvd__0__right,axiom,
    ! [A_208: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_208),zero_zero_nat)) ).

tff(fact_764__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
    ~ ! [S1: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,S1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) ).

tff(fact_765__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
    hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)) = one_one_int ).

tff(fact_766_rel__simps_I47_J,axiom,
    ! [K_1: int] :
      ( ( hAPP_int_int(bit1,K_1) = min )
    <=> ( K_1 = min ) ) ).

tff(fact_767_rel__simps_I43_J,axiom,
    ! [L: int] :
      ( ( min = hAPP_int_int(bit1,L) )
    <=> ( min = L ) ) ).

tff(fact_768_Bit1__Min,axiom,
    hAPP_int_int(bit1,min) = min ).

tff(fact_769_rel__simps_I37_J,axiom,
    pls != min ).

tff(fact_770_rel__simps_I40_J,axiom,
    min != pls ).

tff(fact_771_rel__simps_I45_J,axiom,
    ! [K_2: int] : hAPP_int_int(bit0,K_2) != min ).

tff(fact_772_rel__simps_I42_J,axiom,
    ! [L_2: int] : min != hAPP_int_int(bit0,L_2) ).

tff(fact_773_rel__simps_I7_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).

tff(fact_774_rel__simps_I24_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).

tff(fact_775_rel__simps_I13_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),min)) ) ).

tff(fact_776_rel__simps_I9_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).

tff(fact_777_rel__simps_I3_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).

tff(fact_778_rel__simps_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).

tff(fact_779_rel__simps_I8_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).

tff(fact_780_bin__less__0__simps_I2_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).

tff(fact_781_rel__simps_I26_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K_1)) ) ).

tff(fact_782_rel__simps_I30_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).

tff(fact_783_rel__simps_I23_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).

tff(fact_784_rel__simps_I20_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).

tff(fact_785_rel__simps_I28_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).

tff(fact_786_eq__number__of__Pls__Min,axiom,
    hAPP_int_int(number_number_of_int,pls) != hAPP_int_int(number_number_of_int,min) ).

tff(fact_787_rel__simps_I11_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),min))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) ) ).

tff(fact_788_rel__simps_I25_J,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) ) ).

tff(fact_789_zmult__eq__1__iff,axiom,
    ! [Ma: int,N_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1) = one_one_int )
    <=> ( ( ( Ma = one_one_int )
          & ( N_1 = one_one_int ) )
        | ( ( Ma = hAPP_int_int(number_number_of_int,min) )
          & ( N_1 = hAPP_int_int(number_number_of_int,min) ) ) ) ) ).

tff(fact_790_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_791_linorder__neqE__linordered__idom,axiom,
    ! [X_31: int,Y_25: int] :
      ( ( X_31 != Y_25 )
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_25))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),X_31)) ) ) ).

tff(fact_792_linorder__neqE__linordered__idom,axiom,
    ! [X_31: real,Y_25: real] :
      ( ( X_31 != Y_25 )
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_25))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_25),X_31)) ) ) ).

tff(fact_793_linorder__neqE__linordered__idom,axiom,
    ! [X_31: rat,Y_25: rat] :
      ( ( X_31 != Y_25 )
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_31),Y_25))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_25),X_31)) ) ) ).

tff(fact_794_dvd__refl,axiom,
    ! [A_207: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_207),A_207)) ).

tff(fact_795_dvd__refl,axiom,
    ! [A_207: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_207),A_207)) ).

tff(fact_796_dvd__refl,axiom,
    ! [A_207: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_207),A_207)) ).

tff(fact_797_dvd__refl,axiom,
    ! [A_207: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_207),A_207)) ).

tff(fact_798_dvd__refl,axiom,
    ! [A_207: complex] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_207),A_207)) ).

tff(fact_799_dvd__refl,axiom,
    ! [A_207: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_207),A_207)) ).

tff(fact_800_dvd__refl,axiom,
    ! [A_207: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_207),A_207)) ).

tff(fact_801_dvd__trans,axiom,
    ! [C_88: quickcheck_code_int,A_206: quickcheck_code_int,B_157: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_206),B_157))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_157),C_88))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_206),C_88)) ) ) ).

tff(fact_802_dvd__trans,axiom,
    ! [C_88: code_code_numeral,A_206: code_code_numeral,B_157: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_206),B_157))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_157),C_88))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_206),C_88)) ) ) ).

tff(fact_803_dvd__trans,axiom,
    ! [C_88: rat,A_206: rat,B_157: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_206),B_157))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_157),C_88))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_206),C_88)) ) ) ).

tff(fact_804_dvd__trans,axiom,
    ! [C_88: real,A_206: real,B_157: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_206),B_157))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_157),C_88))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_206),C_88)) ) ) ).

tff(fact_805_dvd__trans,axiom,
    ! [C_88: complex,A_206: complex,B_157: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_206),B_157))
     => ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_157),C_88))
       => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_206),C_88)) ) ) ).

tff(fact_806_dvd__trans,axiom,
    ! [C_88: int,A_206: int,B_157: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_206),B_157))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_157),C_88))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_206),C_88)) ) ) ).

tff(fact_807_dvd__trans,axiom,
    ! [C_88: nat,A_206: nat,B_157: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_206),B_157))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_157),C_88))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_206),C_88)) ) ) ).

tff(fact_808_zcong__sym,axiom,
    ! [A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B),A),Ma)) ) ).

tff(fact_809_zcong__refl,axiom,
    ! [K_2: int,M: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M)) ).

tff(fact_810_zcong__eq__trans,axiom,
    ! [D_3: int,C_1: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => ( ( B_2 = C_1 )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),D_3),M)) ) ) ) ).

tff(fact_811_zcong__trans,axiom,
    ! [C_1: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),C_1),M))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),C_1),M)) ) ) ).

tff(fact_812_zcong__neg__1__impl__ne__1,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(number_number_of_int,min)),P_3))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),one_one_int),P_3)) ) ) ).

tff(fact_813_order__le__neq__implies__less,axiom,
    ! [X_30: code_code_numeral,Y_24: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_30),Y_24))
     => ( ( X_30 != Y_24 )
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_30),Y_24)) ) ) ).

tff(fact_814_order__le__neq__implies__less,axiom,
    ! [X_30: int,Y_24: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_24))
     => ( ( X_30 != Y_24 )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_30),Y_24)) ) ) ).

tff(fact_815_order__le__neq__implies__less,axiom,
    ! [X_30: nat,Y_24: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_24))
     => ( ( X_30 != Y_24 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_30),Y_24)) ) ) ).

tff(fact_816_order__le__neq__implies__less,axiom,
    ! [X_30: real,Y_24: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_30),Y_24))
     => ( ( X_30 != Y_24 )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_30),Y_24)) ) ) ).

tff(fact_817_order__le__neq__implies__less,axiom,
    ! [X_30: quickcheck_code_int,Y_24: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_30),Y_24))
     => ( ( X_30 != Y_24 )
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_30),Y_24)) ) ) ).

tff(fact_818_order__le__neq__implies__less,axiom,
    ! [X_1: fun_nat_bool,Y_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2))
     => ( ( X_1 != Y_2 )
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2)) ) ) ).

tff(fact_819_order__le__neq__implies__less,axiom,
    ! [X_1: fun_int_bool,Y_2: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2))
     => ( ( X_1 != Y_2 )
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2)) ) ) ).

tff(fact_820_order__le__neq__implies__less,axiom,
    ! [X_30: rat,Y_24: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_30),Y_24))
     => ( ( X_30 != Y_24 )
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_30),Y_24)) ) ) ).

tff(fact_821_divisors__zero,axiom,
    ! [A_205: int,B_156: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_205),B_156) = zero_zero_int )
     => ( ( A_205 = zero_zero_int )
        | ( B_156 = zero_zero_int ) ) ) ).

tff(fact_822_divisors__zero,axiom,
    ! [A_205: nat,B_156: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_205),B_156) = zero_zero_nat )
     => ( ( A_205 = zero_zero_nat )
        | ( B_156 = zero_zero_nat ) ) ) ).

tff(fact_823_divisors__zero,axiom,
    ! [A_205: real,B_156: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_205),B_156) = zero_zero_real )
     => ( ( A_205 = zero_zero_real )
        | ( B_156 = zero_zero_real ) ) ) ).

tff(fact_824_divisors__zero,axiom,
    ! [A_205: code_code_numeral,B_156: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_205),B_156) = zero_z126310315umeral )
     => ( ( A_205 = zero_z126310315umeral )
        | ( B_156 = zero_z126310315umeral ) ) ) ).

tff(fact_825_divisors__zero,axiom,
    ! [A_205: complex,B_156: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_205),B_156) = zero_zero_complex )
     => ( ( A_205 = zero_zero_complex )
        | ( B_156 = zero_zero_complex ) ) ) ).

tff(fact_826_divisors__zero,axiom,
    ! [A_205: quickcheck_code_int,B_156: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_205),B_156) = zero_z891286103de_int )
     => ( ( A_205 = zero_z891286103de_int )
        | ( B_156 = zero_z891286103de_int ) ) ) ).

tff(fact_827_divisors__zero,axiom,
    ! [A_205: rat,B_156: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_205),B_156) = zero_zero_rat )
     => ( ( A_205 = zero_zero_rat )
        | ( B_156 = zero_zero_rat ) ) ) ).

tff(fact_828_no__zero__divisors,axiom,
    ! [B_155: int,A_204: int] :
      ( ( A_204 != zero_zero_int )
     => ( ( B_155 != zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_204),B_155) != zero_zero_int ) ) ) ).

tff(fact_829_no__zero__divisors,axiom,
    ! [B_155: nat,A_204: nat] :
      ( ( A_204 != zero_zero_nat )
     => ( ( B_155 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_204),B_155) != zero_zero_nat ) ) ) ).

tff(fact_830_no__zero__divisors,axiom,
    ! [B_155: real,A_204: real] :
      ( ( A_204 != zero_zero_real )
     => ( ( B_155 != zero_zero_real )
       => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_204),B_155) != zero_zero_real ) ) ) ).

tff(fact_831_no__zero__divisors,axiom,
    ! [B_155: code_code_numeral,A_204: code_code_numeral] :
      ( ( A_204 != zero_z126310315umeral )
     => ( ( B_155 != zero_z126310315umeral )
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_204),B_155) != zero_z126310315umeral ) ) ) ).

tff(fact_832_no__zero__divisors,axiom,
    ! [B_155: complex,A_204: complex] :
      ( ( A_204 != zero_zero_complex )
     => ( ( B_155 != zero_zero_complex )
       => ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_204),B_155) != zero_zero_complex ) ) ) ).

tff(fact_833_no__zero__divisors,axiom,
    ! [B_155: quickcheck_code_int,A_204: quickcheck_code_int] :
      ( ( A_204 != zero_z891286103de_int )
     => ( ( B_155 != zero_z891286103de_int )
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_204),B_155) != zero_z891286103de_int ) ) ) ).

tff(fact_834_no__zero__divisors,axiom,
    ! [B_155: rat,A_204: rat] :
      ( ( A_204 != zero_zero_rat )
     => ( ( B_155 != zero_zero_rat )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_204),B_155) != zero_zero_rat ) ) ) ).

tff(fact_835_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

tff(fact_836_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

tff(fact_837_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

tff(fact_838_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

tff(fact_839_mult__zero__right,axiom,
    ! [A_203: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_203),zero_zero_int) = zero_zero_int ).

tff(fact_840_mult__zero__right,axiom,
    ! [A_203: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_203),zero_zero_nat) = zero_zero_nat ).

tff(fact_841_mult__zero__right,axiom,
    ! [A_203: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_203),zero_zero_real) = zero_zero_real ).

tff(fact_842_mult__zero__right,axiom,
    ! [A_203: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_203),zero_z126310315umeral) = zero_z126310315umeral ).

tff(fact_843_mult__zero__right,axiom,
    ! [A_203: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_203),zero_zero_complex) = zero_zero_complex ).

tff(fact_844_mult__zero__right,axiom,
    ! [A_203: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_203),zero_z891286103de_int) = zero_z891286103de_int ).

tff(fact_845_mult__zero__right,axiom,
    ! [A_203: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_203),zero_zero_rat) = zero_zero_rat ).

tff(fact_846_mult__zero__left,axiom,
    ! [A_202: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_202) = zero_zero_int ).

tff(fact_847_mult__zero__left,axiom,
    ! [A_202: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_202) = zero_zero_nat ).

tff(fact_848_mult__zero__left,axiom,
    ! [A_202: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_202) = zero_zero_real ).

tff(fact_849_mult__zero__left,axiom,
    ! [A_202: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_202) = zero_z126310315umeral ).

tff(fact_850_mult__zero__left,axiom,
    ! [A_202: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_202) = zero_zero_complex ).

tff(fact_851_mult__zero__left,axiom,
    ! [A_202: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_202) = zero_z891286103de_int ).

tff(fact_852_mult__zero__left,axiom,
    ! [A_202: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_202) = zero_zero_rat ).

tff(fact_853_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

tff(fact_854_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

tff(fact_855_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

tff(fact_856_zero__neq__one,axiom,
    zero_z126310315umeral != one_on1645066479umeral ).

tff(fact_857_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

tff(fact_858_zero__neq__one,axiom,
    zero_z891286103de_int != one_on1684967323de_int ).

tff(fact_859_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

tff(fact_860_one__neq__zero,axiom,
    one_one_int != zero_zero_int ).

tff(fact_861_one__neq__zero,axiom,
    one_one_nat != zero_zero_nat ).

tff(fact_862_one__neq__zero,axiom,
    one_one_real != zero_zero_real ).

tff(fact_863_one__neq__zero,axiom,
    one_on1645066479umeral != zero_z126310315umeral ).

tff(fact_864_one__neq__zero,axiom,
    one_one_complex != zero_zero_complex ).

tff(fact_865_one__neq__zero,axiom,
    one_on1684967323de_int != zero_z891286103de_int ).

tff(fact_866_one__neq__zero,axiom,
    one_one_rat != zero_zero_rat ).

tff(fact_867_combine__common__factor,axiom,
    ! [A_201: code_code_numeral,E_3: code_code_numeral,B_154: code_code_numeral,C_87: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_201),E_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_154),E_3)),C_87)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_201),B_154)),E_3)),C_87) ).

tff(fact_868_combine__common__factor,axiom,
    ! [A_201: int,E_3: int,B_154: int,C_87: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_201),E_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_154),E_3)),C_87)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_201),B_154)),E_3)),C_87) ).

tff(fact_869_combine__common__factor,axiom,
    ! [A_201: nat,E_3: nat,B_154: nat,C_87: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_201),E_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_154),E_3)),C_87)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_201),B_154)),E_3)),C_87) ).

tff(fact_870_combine__common__factor,axiom,
    ! [A_201: real,E_3: real,B_154: real,C_87: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_201),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_154),E_3)),C_87)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_201),B_154)),E_3)),C_87) ).

tff(fact_871_combine__common__factor,axiom,
    ! [A_201: complex,E_3: complex,B_154: complex,C_87: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_201),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_154),E_3)),C_87)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_201),B_154)),E_3)),C_87) ).

tff(fact_872_combine__common__factor,axiom,
    ! [A_201: quickcheck_code_int,E_3: quickcheck_code_int,B_154: quickcheck_code_int,C_87: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_201),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_154),E_3)),C_87)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_201),B_154)),E_3)),C_87) ).

tff(fact_873_combine__common__factor,axiom,
    ! [A_201: rat,E_3: rat,B_154: rat,C_87: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_201),E_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_154),E_3)),C_87)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_201),B_154)),E_3)),C_87) ).

tff(fact_874_comm__semiring__class_Odistrib,axiom,
    ! [A_200: code_code_numeral,B_153: code_code_numeral,C_86: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_200),B_153)),C_86) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_200),C_86)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_153),C_86)) ).

tff(fact_875_comm__semiring__class_Odistrib,axiom,
    ! [A_200: int,B_153: int,C_86: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_200),B_153)),C_86) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_200),C_86)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_153),C_86)) ).

tff(fact_876_comm__semiring__class_Odistrib,axiom,
    ! [A_200: nat,B_153: nat,C_86: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_200),B_153)),C_86) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_200),C_86)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_153),C_86)) ).

tff(fact_877_comm__semiring__class_Odistrib,axiom,
    ! [A_200: real,B_153: real,C_86: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_200),B_153)),C_86) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_200),C_86)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_153),C_86)) ).

tff(fact_878_comm__semiring__class_Odistrib,axiom,
    ! [A_200: complex,B_153: complex,C_86: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_200),B_153)),C_86) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_200),C_86)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_153),C_86)) ).

tff(fact_879_comm__semiring__class_Odistrib,axiom,
    ! [A_200: quickcheck_code_int,B_153: quickcheck_code_int,C_86: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_200),B_153)),C_86) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_200),C_86)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_153),C_86)) ).

tff(fact_880_comm__semiring__class_Odistrib,axiom,
    ! [A_200: rat,B_153: rat,C_86: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_200),B_153)),C_86) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_200),C_86)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_153),C_86)) ).

tff(fact_881_field__power__not__zero,axiom,
    ! [N_65: nat,A_199: int] :
      ( ( A_199 != zero_zero_int )
     => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_65) != zero_zero_int ) ) ).

tff(fact_882_field__power__not__zero,axiom,
    ! [N_65: nat,A_199: real] :
      ( ( A_199 != zero_zero_real )
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_65) != zero_zero_real ) ) ).

tff(fact_883_field__power__not__zero,axiom,
    ! [N_65: nat,A_199: complex] :
      ( ( A_199 != zero_zero_complex )
     => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_199),N_65) != zero_zero_complex ) ) ).

tff(fact_884_field__power__not__zero,axiom,
    ! [N_65: nat,A_199: rat] :
      ( ( A_199 != zero_zero_rat )
     => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_65) != zero_zero_rat ) ) ).

tff(fact_885_dvd__0__left,axiom,
    ! [A_198: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_198))
     => ( A_198 = zero_zero_real ) ) ).

tff(fact_886_dvd__0__left,axiom,
    ! [A_198: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,zero_z126310315umeral),A_198))
     => ( A_198 = zero_z126310315umeral ) ) ).

tff(fact_887_dvd__0__left,axiom,
    ! [A_198: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,zero_zero_complex),A_198))
     => ( A_198 = zero_zero_complex ) ) ).

tff(fact_888_dvd__0__left,axiom,
    ! [A_198: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,zero_z891286103de_int),A_198))
     => ( A_198 = zero_z891286103de_int ) ) ).

tff(fact_889_dvd__0__left,axiom,
    ! [A_198: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,zero_zero_rat),A_198))
     => ( A_198 = zero_zero_rat ) ) ).

tff(fact_890_dvd__0__left,axiom,
    ! [A_198: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_198))
     => ( A_198 = zero_zero_int ) ) ).

tff(fact_891_dvd__0__left,axiom,
    ! [A_198: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_198))
     => ( A_198 = zero_zero_nat ) ) ).

tff(fact_892_power__mult__distrib,axiom,
    ! [A_197: code_code_numeral,B_152: code_code_numeral,N_64: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_197),B_152)),N_64) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_197),N_64)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_152),N_64)) ).

tff(fact_893_power__mult__distrib,axiom,
    ! [A_197: quickcheck_code_int,B_152: quickcheck_code_int,N_64: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_197),B_152)),N_64) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_197),N_64)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_152),N_64)) ).

tff(fact_894_power__mult__distrib,axiom,
    ! [A_197: rat,B_152: rat,N_64: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_197),B_152)),N_64) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),N_64)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_152),N_64)) ).

tff(fact_895_power__mult__distrib,axiom,
    ! [A_197: int,B_152: int,N_64: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_197),B_152)),N_64) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_197),N_64)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_152),N_64)) ).

tff(fact_896_power__mult__distrib,axiom,
    ! [A_197: nat,B_152: nat,N_64: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_197),B_152)),N_64) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_197),N_64)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_152),N_64)) ).

tff(fact_897_power__mult__distrib,axiom,
    ! [A_197: real,B_152: real,N_64: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_197),B_152)),N_64) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_197),N_64)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_152),N_64)) ).

tff(fact_898_power__mult__distrib,axiom,
    ! [A_197: complex,B_152: complex,N_64: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_197),B_152)),N_64) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_197),N_64)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,B_152),N_64)) ).

tff(fact_899_power__commutes,axiom,
    ! [A_196: code_code_numeral,N_63: nat] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_196),N_63)),A_196) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_196),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_196),N_63)) ).

tff(fact_900_power__commutes,axiom,
    ! [A_196: quickcheck_code_int,N_63: nat] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_196),N_63)),A_196) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_196),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_196),N_63)) ).

tff(fact_901_power__commutes,axiom,
    ! [A_196: rat,N_63: nat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_196),N_63)),A_196) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_196),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_196),N_63)) ).

tff(fact_902_power__commutes,axiom,
    ! [A_196: int,N_63: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_196),N_63)),A_196) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_196),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_196),N_63)) ).

tff(fact_903_power__commutes,axiom,
    ! [A_196: nat,N_63: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_196),N_63)),A_196) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_196),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_196),N_63)) ).

tff(fact_904_power__commutes,axiom,
    ! [A_196: real,N_63: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_196),N_63)),A_196) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_196),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_196),N_63)) ).

tff(fact_905_power__commutes,axiom,
    ! [A_196: complex,N_63: nat] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_196),N_63)),A_196) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_196),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_196),N_63)) ).

tff(fact_906_dvd__triv__left,axiom,
    ! [A_195: code_code_numeral,B_151: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_195),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_195),B_151))) ).

tff(fact_907_dvd__triv__left,axiom,
    ! [A_195: real,B_151: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_195),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_195),B_151))) ).

tff(fact_908_dvd__triv__left,axiom,
    ! [A_195: complex,B_151: complex] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_195),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_195),B_151))) ).

tff(fact_909_dvd__triv__left,axiom,
    ! [A_195: quickcheck_code_int,B_151: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_195),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_195),B_151))) ).

tff(fact_910_dvd__triv__left,axiom,
    ! [A_195: rat,B_151: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_195),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_195),B_151))) ).

tff(fact_911_dvd__triv__left,axiom,
    ! [A_195: int,B_151: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_195),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_195),B_151))) ).

tff(fact_912_dvd__triv__left,axiom,
    ! [A_195: nat,B_151: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_195),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_195),B_151))) ).

tff(fact_913_dvd__triv__right,axiom,
    ! [A_194: code_code_numeral,B_150: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_194),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_150),A_194))) ).

tff(fact_914_dvd__triv__right,axiom,
    ! [A_194: real,B_150: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_194),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_150),A_194))) ).

tff(fact_915_dvd__triv__right,axiom,
    ! [A_194: complex,B_150: complex] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_194),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_150),A_194))) ).

tff(fact_916_dvd__triv__right,axiom,
    ! [A_194: quickcheck_code_int,B_150: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_194),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_150),A_194))) ).

tff(fact_917_dvd__triv__right,axiom,
    ! [A_194: rat,B_150: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_194),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_150),A_194))) ).

tff(fact_918_dvd__triv__right,axiom,
    ! [A_194: int,B_150: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_194),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_150),A_194))) ).

tff(fact_919_dvd__triv__right,axiom,
    ! [A_194: nat,B_150: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_194),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_150),A_194))) ).

tff(fact_920_dvd__mult2,axiom,
    ! [C_85: code_code_numeral,A_193: code_code_numeral,B_149: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_193),B_149))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_193),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_149),C_85))) ) ).

tff(fact_921_dvd__mult2,axiom,
    ! [C_85: real,A_193: real,B_149: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_193),B_149))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_193),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_149),C_85))) ) ).

tff(fact_922_dvd__mult2,axiom,
    ! [C_85: complex,A_193: complex,B_149: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_193),B_149))
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_193),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_149),C_85))) ) ).

tff(fact_923_dvd__mult2,axiom,
    ! [C_85: quickcheck_code_int,A_193: quickcheck_code_int,B_149: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_193),B_149))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_193),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_149),C_85))) ) ).

tff(fact_924_dvd__mult2,axiom,
    ! [C_85: rat,A_193: rat,B_149: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_193),B_149))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_193),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_149),C_85))) ) ).

tff(fact_925_dvd__mult2,axiom,
    ! [C_85: int,A_193: int,B_149: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_193),B_149))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_193),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_149),C_85))) ) ).

tff(fact_926_dvd__mult2,axiom,
    ! [C_85: nat,A_193: nat,B_149: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_193),B_149))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_149),C_85))) ) ).

tff(fact_927_dvd__mult,axiom,
    ! [B_148: code_code_numeral,A_192: code_code_numeral,C_84: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_192),C_84))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_192),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_148),C_84))) ) ).

tff(fact_928_dvd__mult,axiom,
    ! [B_148: real,A_192: real,C_84: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_192),C_84))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_192),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_148),C_84))) ) ).

tff(fact_929_dvd__mult,axiom,
    ! [B_148: complex,A_192: complex,C_84: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_192),C_84))
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_192),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_148),C_84))) ) ).

tff(fact_930_dvd__mult,axiom,
    ! [B_148: quickcheck_code_int,A_192: quickcheck_code_int,C_84: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_192),C_84))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_192),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_148),C_84))) ) ).

tff(fact_931_dvd__mult,axiom,
    ! [B_148: rat,A_192: rat,C_84: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_192),C_84))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_192),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_148),C_84))) ) ).

tff(fact_932_dvd__mult,axiom,
    ! [B_148: int,A_192: int,C_84: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_192),C_84))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_192),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_148),C_84))) ) ).

tff(fact_933_dvd__mult,axiom,
    ! [B_148: nat,A_192: nat,C_84: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_192),C_84))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_148),C_84))) ) ).

tff(fact_934_mult__dvd__mono,axiom,
    ! [C_83: code_code_numeral,D_16: code_code_numeral,A_191: code_code_numeral,B_147: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_191),B_147))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_83),D_16))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_191),C_83)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_147),D_16))) ) ) ).

tff(fact_935_mult__dvd__mono,axiom,
    ! [C_83: real,D_16: real,A_191: real,B_147: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_191),B_147))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_83),D_16))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_191),C_83)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_147),D_16))) ) ) ).

tff(fact_936_mult__dvd__mono,axiom,
    ! [C_83: complex,D_16: complex,A_191: complex,B_147: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_191),B_147))
     => ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,C_83),D_16))
       => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_191),C_83)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_147),D_16))) ) ) ).

tff(fact_937_mult__dvd__mono,axiom,
    ! [C_83: quickcheck_code_int,D_16: quickcheck_code_int,A_191: quickcheck_code_int,B_147: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_191),B_147))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_83),D_16))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_191),C_83)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_147),D_16))) ) ) ).

tff(fact_938_mult__dvd__mono,axiom,
    ! [C_83: rat,D_16: rat,A_191: rat,B_147: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_191),B_147))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,C_83),D_16))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_191),C_83)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_147),D_16))) ) ) ).

tff(fact_939_mult__dvd__mono,axiom,
    ! [C_83: int,D_16: int,A_191: int,B_147: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_191),B_147))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_83),D_16))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_191),C_83)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_147),D_16))) ) ) ).

tff(fact_940_mult__dvd__mono,axiom,
    ! [C_83: nat,D_16: nat,A_191: nat,B_147: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_191),B_147))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_83),D_16))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_191),C_83)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_147),D_16))) ) ) ).

tff(fact_941_dvdI,axiom,
    ! [A_190: code_code_numeral,B_146: code_code_numeral,K_7: code_code_numeral] :
      ( ( A_190 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_146),K_7) )
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_146),A_190)) ) ).

tff(fact_942_dvdI,axiom,
    ! [A_190: real,B_146: real,K_7: real] :
      ( ( A_190 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_146),K_7) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_146),A_190)) ) ).

tff(fact_943_dvdI,axiom,
    ! [A_190: complex,B_146: complex,K_7: complex] :
      ( ( A_190 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_146),K_7) )
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_146),A_190)) ) ).

tff(fact_944_dvdI,axiom,
    ! [A_190: quickcheck_code_int,B_146: quickcheck_code_int,K_7: quickcheck_code_int] :
      ( ( A_190 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_146),K_7) )
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_146),A_190)) ) ).

tff(fact_945_dvdI,axiom,
    ! [A_190: rat,B_146: rat,K_7: rat] :
      ( ( A_190 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_146),K_7) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_146),A_190)) ) ).

tff(fact_946_dvdI,axiom,
    ! [A_190: int,B_146: int,K_7: int] :
      ( ( A_190 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_146),K_7) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_146),A_190)) ) ).

tff(fact_947_dvdI,axiom,
    ! [A_190: nat,B_146: nat,K_7: nat] :
      ( ( A_190 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_146),K_7) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_146),A_190)) ) ).

tff(fact_948_dvd__mult__left,axiom,
    ! [A_189: code_code_numeral,B_145: code_code_numeral,C_82: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_189),B_145)),C_82))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_189),C_82)) ) ).

tff(fact_949_dvd__mult__left,axiom,
    ! [A_189: real,B_145: real,C_82: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_189),B_145)),C_82))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_189),C_82)) ) ).

tff(fact_950_dvd__mult__left,axiom,
    ! [A_189: complex,B_145: complex,C_82: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_189),B_145)),C_82))
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_189),C_82)) ) ).

tff(fact_951_dvd__mult__left,axiom,
    ! [A_189: quickcheck_code_int,B_145: quickcheck_code_int,C_82: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_189),B_145)),C_82))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_189),C_82)) ) ).

tff(fact_952_dvd__mult__left,axiom,
    ! [A_189: rat,B_145: rat,C_82: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_189),B_145)),C_82))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_189),C_82)) ) ).

tff(fact_953_dvd__mult__left,axiom,
    ! [A_189: int,B_145: int,C_82: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_189),B_145)),C_82))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_189),C_82)) ) ).

tff(fact_954_dvd__mult__left,axiom,
    ! [A_189: nat,B_145: nat,C_82: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_189),B_145)),C_82))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_189),C_82)) ) ).

tff(fact_955_dvd__mult__right,axiom,
    ! [A_188: code_code_numeral,B_144: code_code_numeral,C_81: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_188),B_144)),C_81))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_144),C_81)) ) ).

tff(fact_956_dvd__mult__right,axiom,
    ! [A_188: real,B_144: real,C_81: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_188),B_144)),C_81))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_144),C_81)) ) ).

tff(fact_957_dvd__mult__right,axiom,
    ! [A_188: complex,B_144: complex,C_81: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_188),B_144)),C_81))
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_144),C_81)) ) ).

tff(fact_958_dvd__mult__right,axiom,
    ! [A_188: quickcheck_code_int,B_144: quickcheck_code_int,C_81: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_188),B_144)),C_81))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_144),C_81)) ) ).

tff(fact_959_dvd__mult__right,axiom,
    ! [A_188: rat,B_144: rat,C_81: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_188),B_144)),C_81))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_144),C_81)) ) ).

tff(fact_960_dvd__mult__right,axiom,
    ! [A_188: int,B_144: int,C_81: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_188),B_144)),C_81))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_144),C_81)) ) ).

tff(fact_961_dvd__mult__right,axiom,
    ! [A_188: nat,B_144: nat,C_81: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_188),B_144)),C_81))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_144),C_81)) ) ).

tff(fact_962_dvd__add,axiom,
    ! [C_80: code_code_numeral,A_187: code_code_numeral,B_143: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_187),B_143))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_187),C_80))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_187),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_143),C_80))) ) ) ).

tff(fact_963_dvd__add,axiom,
    ! [C_80: real,A_187: real,B_143: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_187),B_143))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_187),C_80))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_187),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_143),C_80))) ) ) ).

tff(fact_964_dvd__add,axiom,
    ! [C_80: complex,A_187: complex,B_143: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_187),B_143))
     => ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_187),C_80))
       => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_187),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_143),C_80))) ) ) ).

tff(fact_965_dvd__add,axiom,
    ! [C_80: quickcheck_code_int,A_187: quickcheck_code_int,B_143: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_187),B_143))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_187),C_80))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_187),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_143),C_80))) ) ) ).

tff(fact_966_dvd__add,axiom,
    ! [C_80: rat,A_187: rat,B_143: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_187),B_143))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_187),C_80))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_187),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_143),C_80))) ) ) ).

tff(fact_967_dvd__add,axiom,
    ! [C_80: int,A_187: int,B_143: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_187),B_143))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_187),C_80))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_187),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_143),C_80))) ) ) ).

tff(fact_968_dvd__add,axiom,
    ! [C_80: nat,A_187: nat,B_143: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_187),B_143))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_187),C_80))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_187),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_143),C_80))) ) ) ).

tff(fact_969_power__one,axiom,
    ! [N_62: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),N_62) = one_one_int ).

tff(fact_970_power__one,axiom,
    ! [N_62: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),N_62) = one_one_nat ).

tff(fact_971_power__one,axiom,
    ! [N_62: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),N_62) = one_one_real ).

tff(fact_972_power__one,axiom,
    ! [N_62: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),N_62) = one_on1645066479umeral ).

tff(fact_973_power__one,axiom,
    ! [N_62: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),N_62) = one_one_complex ).

tff(fact_974_power__one,axiom,
    ! [N_62: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_62) = one_on1684967323de_int ).

tff(fact_975_power__one,axiom,
    ! [N_62: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),N_62) = one_one_rat ).

tff(fact_976_one__dvd,axiom,
    ! [A_186: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_186)) ).

tff(fact_977_one__dvd,axiom,
    ! [A_186: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,one_on1645066479umeral),A_186)) ).

tff(fact_978_one__dvd,axiom,
    ! [A_186: complex] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,one_one_complex),A_186)) ).

tff(fact_979_one__dvd,axiom,
    ! [A_186: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,one_on1684967323de_int),A_186)) ).

tff(fact_980_one__dvd,axiom,
    ! [A_186: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,one_one_rat),A_186)) ).

tff(fact_981_one__dvd,axiom,
    ! [A_186: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_186)) ).

tff(fact_982_one__dvd,axiom,
    ! [A_186: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_186)) ).

tff(fact_983_dvd__power__same,axiom,
    ! [N_61: nat,X_29: quickcheck_code_int,Y_23: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_29),Y_23))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_29),N_61)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_23),N_61))) ) ).

tff(fact_984_dvd__power__same,axiom,
    ! [N_61: nat,X_29: code_code_numeral,Y_23: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_29),Y_23))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_29),N_61)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_23),N_61))) ) ).

tff(fact_985_dvd__power__same,axiom,
    ! [N_61: nat,X_29: rat,Y_23: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_29),Y_23))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_29),N_61)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_23),N_61))) ) ).

tff(fact_986_dvd__power__same,axiom,
    ! [N_61: nat,X_29: real,Y_23: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_29),Y_23))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_29),N_61)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_23),N_61))) ) ).

tff(fact_987_dvd__power__same,axiom,
    ! [N_61: nat,X_29: complex,Y_23: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_29),Y_23))
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_29),N_61)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_23),N_61))) ) ).

tff(fact_988_dvd__power__same,axiom,
    ! [N_61: nat,X_29: int,Y_23: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_29),Y_23))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_29),N_61)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_23),N_61))) ) ).

tff(fact_989_dvd__power__same,axiom,
    ! [N_61: nat,X_29: nat,Y_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_29),Y_23))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_29),N_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_23),N_61))) ) ).

tff(fact_990_power__le__dvd,axiom,
    ! [M_25: nat,A_185: quickcheck_code_int,N_60: nat,B_142: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_185),M_25)),B_142)) ) ) ).

tff(fact_991_power__le__dvd,axiom,
    ! [M_25: nat,A_185: code_code_numeral,N_60: nat,B_142: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_185),M_25)),B_142)) ) ) ).

tff(fact_992_power__le__dvd,axiom,
    ! [M_25: nat,A_185: rat,N_60: nat,B_142: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_185),M_25)),B_142)) ) ) ).

tff(fact_993_power__le__dvd,axiom,
    ! [M_25: nat,A_185: real,N_60: nat,B_142: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_185),M_25)),B_142)) ) ) ).

tff(fact_994_power__le__dvd,axiom,
    ! [M_25: nat,A_185: complex,N_60: nat,B_142: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_185),M_25)),B_142)) ) ) ).

tff(fact_995_power__le__dvd,axiom,
    ! [M_25: nat,A_185: int,N_60: nat,B_142: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_185),M_25)),B_142)) ) ) ).

tff(fact_996_power__le__dvd,axiom,
    ! [M_25: nat,A_185: nat,N_60: nat,B_142: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_185),N_60)),B_142))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_185),M_25)),B_142)) ) ) ).

tff(fact_997_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: quickcheck_code_int,Y_22: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_28),N_59)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_22),M_24))) ) ) ).

tff(fact_998_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: code_code_numeral,Y_22: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_28),N_59)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_22),M_24))) ) ) ).

tff(fact_999_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: rat,Y_22: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_28),N_59)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_22),M_24))) ) ) ).

tff(fact_1000_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: real,Y_22: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_28),N_59)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_22),M_24))) ) ) ).

tff(fact_1001_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: complex,Y_22: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_28),N_59)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_22),M_24))) ) ) ).

tff(fact_1002_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: int,Y_22: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_28),N_59)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_22),M_24))) ) ) ).

tff(fact_1003_dvd__power__le,axiom,
    ! [N_59: nat,M_24: nat,X_28: nat,Y_22: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_28),Y_22))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_28),N_59)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_22),M_24))) ) ) ).

tff(fact_1004_le__imp__power__dvd,axiom,
    ! [A_184: quickcheck_code_int,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_184),M_23)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_184),N_58))) ) ).

tff(fact_1005_le__imp__power__dvd,axiom,
    ! [A_184: code_code_numeral,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_184),M_23)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_184),N_58))) ) ).

tff(fact_1006_le__imp__power__dvd,axiom,
    ! [A_184: rat,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_184),M_23)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_184),N_58))) ) ).

tff(fact_1007_le__imp__power__dvd,axiom,
    ! [A_184: real,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_184),M_23)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_184),N_58))) ) ).

tff(fact_1008_le__imp__power__dvd,axiom,
    ! [A_184: complex,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_184),M_23)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_184),N_58))) ) ).

tff(fact_1009_le__imp__power__dvd,axiom,
    ! [A_184: int,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_184),M_23)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_184),N_58))) ) ).

tff(fact_1010_le__imp__power__dvd,axiom,
    ! [A_184: nat,M_23: nat,N_58: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_184),M_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_184),N_58))) ) ).

tff(fact_1011_nat__power__less__imp__less,axiom,
    ! [M: nat,N: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_1012_nat__zero__less__power__iff,axiom,
    ! [X_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1))
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_1013_power__mult,axiom,
    ! [A_183: quickcheck_code_int,M_22: nat,N_57: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),M_22)),N_57) ).

tff(fact_1014_power__mult,axiom,
    ! [A_183: code_code_numeral,M_22: nat,N_57: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),M_22)),N_57) ).

tff(fact_1015_power__mult,axiom,
    ! [A_183: rat,M_22: nat,N_57: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),M_22)),N_57) ).

tff(fact_1016_power__mult,axiom,
    ! [A_183: int,M_22: nat,N_57: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),M_22)),N_57) ).

tff(fact_1017_power__mult,axiom,
    ! [A_183: nat,M_22: nat,N_57: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),M_22)),N_57) ).

tff(fact_1018_power__mult,axiom,
    ! [A_183: real,M_22: nat,N_57: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),M_22)),N_57) ).

tff(fact_1019_power__mult,axiom,
    ! [A_183: complex,M_22: nat,N_57: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_183),M_22)),N_57) ).

tff(fact_1020_Euler_Oaux2,axiom,
    ! [B_2: int,A_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),C_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),B_2))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_3)) ) ) ) ).

tff(fact_1021_power__one__right,axiom,
    ! [A_182: quickcheck_code_int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_182),one_one_nat) = A_182 ).

tff(fact_1022_power__one__right,axiom,
    ! [A_182: code_code_numeral] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_182),one_one_nat) = A_182 ).

tff(fact_1023_power__one__right,axiom,
    ! [A_182: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_182),one_one_nat) = A_182 ).

tff(fact_1024_power__one__right,axiom,
    ! [A_182: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_182),one_one_nat) = A_182 ).

tff(fact_1025_power__one__right,axiom,
    ! [A_182: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_182),one_one_nat) = A_182 ).

tff(fact_1026_power__one__right,axiom,
    ! [A_182: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_182),one_one_nat) = A_182 ).

tff(fact_1027_power__one__right,axiom,
    ! [A_182: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_182),one_one_nat) = A_182 ).

tff(fact_1028_zcong__id,axiom,
    ! [M: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M),zero_zero_int),M)) ).

tff(fact_1029_IntPrimes_Ozcong__zero,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),zero_zero_int))
    <=> ( A = B ) ) ).

tff(fact_1030_zcong__1,axiom,
    ! [A_3: int,B_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),one_one_int)) ).

tff(fact_1031_zcong__zmult__self,axiom,
    ! [A_3: int,M: int,B_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),M)) ).

tff(fact_1032_zcong__zmult__prop1,axiom,
    ! [C: int,D: int,A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),Ma))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)),Ma)) ) ) ).

tff(fact_1033_zcong__zmult__prop2,axiom,
    ! [C: int,D: int,A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),A)),Ma))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),B)),Ma)) ) ) ).

tff(fact_1034_zcong__scalar,axiom,
    ! [K_2: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),K_2)),M)) ) ).

tff(fact_1035_zcong__scalar2,axiom,
    ! [K_2: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),B_2)),M)) ) ).

tff(fact_1036_zcong__zmult,axiom,
    ! [C_1: int,D_3: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)),M)) ) ) ).

tff(fact_1037_zcong__zadd,axiom,
    ! [C_1: int,D_3: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),D_3)),M)) ) ) ).

tff(fact_1038_zcong__shift,axiom,
    ! [C_1: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),C_1)),M)) ) ).

tff(fact_1039_zcong__zpower,axiom,
    ! [Z_1: nat,X: int,Y: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),Z_1)),M)) ) ).

tff(fact_1040_power__m1__even,axiom,
    ! [N_56: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56)) = one_one_int ).

tff(fact_1041_power__m1__even,axiom,
    ! [N_56: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56)) = one_one_real ).

tff(fact_1042_power__m1__even,axiom,
    ! [N_56: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56)) = one_one_complex ).

tff(fact_1043_power__m1__even,axiom,
    ! [N_56: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56)) = one_one_rat ).

tff(fact_1044_split__mult__neg__le,axiom,
    ! [B_141: int,A_181: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_181))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_141),zero_zero_int)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_181),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_141)) ) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_181),B_141)),zero_zero_int)) ) ).

tff(fact_1045_split__mult__neg__le,axiom,
    ! [B_141: nat,A_181: nat] :
      ( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_181))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_141),zero_zero_nat)) )
        | ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_181),zero_zero_nat))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_141)) ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_181),B_141)),zero_zero_nat)) ) ).

tff(fact_1046_split__mult__neg__le,axiom,
    ! [B_141: real,A_181: real] :
      ( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_181))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_141),zero_zero_real)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_181),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_141)) ) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_181),B_141)),zero_zero_real)) ) ).

tff(fact_1047_split__mult__neg__le,axiom,
    ! [B_141: code_code_numeral,A_181: code_code_numeral] :
      ( ( ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_181))
          & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_141),zero_z126310315umeral)) )
        | ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_181),zero_z126310315umeral))
          & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_141)) ) )
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_181),B_141)),zero_z126310315umeral)) ) ).

tff(fact_1048_split__mult__neg__le,axiom,
    ! [B_141: quickcheck_code_int,A_181: quickcheck_code_int] :
      ( ( ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_181))
          & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_141),zero_z891286103de_int)) )
        | ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_181),zero_z891286103de_int))
          & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_141)) ) )
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_181),B_141)),zero_z891286103de_int)) ) ).

tff(fact_1049_split__mult__neg__le,axiom,
    ! [B_141: rat,A_181: rat] :
      ( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_181))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_141),zero_zero_rat)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_181),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_141)) ) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_181),B_141)),zero_zero_rat)) ) ).

tff(fact_1050_split__mult__pos__le,axiom,
    ! [B_140: int,A_180: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_180))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_140)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_180),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_140),zero_zero_int)) ) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_180),B_140))) ) ).

tff(fact_1051_split__mult__pos__le,axiom,
    ! [B_140: real,A_180: real] :
      ( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_180))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_140)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_180),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_140),zero_zero_real)) ) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_180),B_140))) ) ).

tff(fact_1052_split__mult__pos__le,axiom,
    ! [B_140: rat,A_180: rat] :
      ( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_180))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_140)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_180),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_140),zero_zero_rat)) ) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_180),B_140))) ) ).

tff(fact_1053_mult__mono,axiom,
    ! [C_79: int,D_15: int,A_179: int,B_139: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_179),B_139))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_79),D_15))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_139))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_79))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_179),C_79)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_139),D_15))) ) ) ) ) ).

tff(fact_1054_mult__mono,axiom,
    ! [C_79: nat,D_15: nat,A_179: nat,B_139: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_179),B_139))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_79),D_15))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_139))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_79))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_179),C_79)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_139),D_15))) ) ) ) ) ).

tff(fact_1055_mult__mono,axiom,
    ! [C_79: real,D_15: real,A_179: real,B_139: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_179),B_139))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_79),D_15))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_139))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_79))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_179),C_79)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_139),D_15))) ) ) ) ) ).

tff(fact_1056_mult__mono,axiom,
    ! [C_79: code_code_numeral,D_15: code_code_numeral,A_179: code_code_numeral,B_139: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_179),B_139))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_79),D_15))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_139))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_79))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_179),C_79)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_139),D_15))) ) ) ) ) ).

tff(fact_1057_mult__mono,axiom,
    ! [C_79: quickcheck_code_int,D_15: quickcheck_code_int,A_179: quickcheck_code_int,B_139: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_179),B_139))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_79),D_15))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_139))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_79))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_179),C_79)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_139),D_15))) ) ) ) ) ).

tff(fact_1058_mult__mono,axiom,
    ! [C_79: rat,D_15: rat,A_179: rat,B_139: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_179),B_139))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_79),D_15))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_139))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_79))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_179),C_79)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_139),D_15))) ) ) ) ) ).

tff(fact_1059_mult__mono_H,axiom,
    ! [C_78: int,D_14: int,A_178: int,B_138: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_178),B_138))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_78),D_14))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_178))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_78))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_178),C_78)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_138),D_14))) ) ) ) ) ).

tff(fact_1060_mult__mono_H,axiom,
    ! [C_78: nat,D_14: nat,A_178: nat,B_138: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_178),B_138))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_78),D_14))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_178))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_78))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_178),C_78)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_138),D_14))) ) ) ) ) ).

tff(fact_1061_mult__mono_H,axiom,
    ! [C_78: real,D_14: real,A_178: real,B_138: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_178),B_138))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_78),D_14))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_178))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_78))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_178),C_78)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_138),D_14))) ) ) ) ) ).

tff(fact_1062_mult__mono_H,axiom,
    ! [C_78: code_code_numeral,D_14: code_code_numeral,A_178: code_code_numeral,B_138: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_178),B_138))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_78),D_14))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_178))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_78))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_178),C_78)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_138),D_14))) ) ) ) ) ).

tff(fact_1063_mult__mono_H,axiom,
    ! [C_78: quickcheck_code_int,D_14: quickcheck_code_int,A_178: quickcheck_code_int,B_138: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_178),B_138))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_78),D_14))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_178))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_78))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_178),C_78)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_138),D_14))) ) ) ) ) ).

tff(fact_1064_mult__mono_H,axiom,
    ! [C_78: rat,D_14: rat,A_178: rat,B_138: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_178),B_138))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_78),D_14))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_178))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_78))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_178),C_78)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_138),D_14))) ) ) ) ) ).

tff(fact_1065_mult__left__mono__neg,axiom,
    ! [C_77: int,B_137: int,A_177: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_137),A_177))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_77),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),A_177)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),B_137))) ) ) ).

tff(fact_1066_mult__left__mono__neg,axiom,
    ! [C_77: real,B_137: real,A_177: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_137),A_177))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_77),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),A_177)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),B_137))) ) ) ).

tff(fact_1067_mult__left__mono__neg,axiom,
    ! [C_77: rat,B_137: rat,A_177: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_137),A_177))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_77),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),A_177)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),B_137))) ) ) ).

tff(fact_1068_mult__right__mono__neg,axiom,
    ! [C_76: int,B_136: int,A_176: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_136),A_176))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_76),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_176),C_76)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_136),C_76))) ) ) ).

tff(fact_1069_mult__right__mono__neg,axiom,
    ! [C_76: real,B_136: real,A_176: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_136),A_176))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_76),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_176),C_76)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_136),C_76))) ) ) ).

tff(fact_1070_mult__right__mono__neg,axiom,
    ! [C_76: rat,B_136: rat,A_176: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_136),A_176))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_76),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_176),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_136),C_76))) ) ) ).

tff(fact_1071_comm__mult__left__mono,axiom,
    ! [C_75: int,A_175: int,B_135: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_175),B_135))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_75))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),A_175)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),B_135))) ) ) ).

tff(fact_1072_comm__mult__left__mono,axiom,
    ! [C_75: nat,A_175: nat,B_135: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_175),B_135))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_75))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),A_175)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),B_135))) ) ) ).

tff(fact_1073_comm__mult__left__mono,axiom,
    ! [C_75: real,A_175: real,B_135: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_175),B_135))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_75))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),A_175)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),B_135))) ) ) ).

tff(fact_1074_comm__mult__left__mono,axiom,
    ! [C_75: code_code_numeral,A_175: code_code_numeral,B_135: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_175),B_135))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_75))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),A_175)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),B_135))) ) ) ).

tff(fact_1075_comm__mult__left__mono,axiom,
    ! [C_75: quickcheck_code_int,A_175: quickcheck_code_int,B_135: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_175),B_135))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_75))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),A_175)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),B_135))) ) ) ).

tff(fact_1076_comm__mult__left__mono,axiom,
    ! [C_75: rat,A_175: rat,B_135: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_175),B_135))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_75))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),A_175)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),B_135))) ) ) ).

tff(fact_1077_mult__left__mono,axiom,
    ! [C_74: int,A_174: int,B_134: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_174),B_134))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_74))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),A_174)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),B_134))) ) ) ).

tff(fact_1078_mult__left__mono,axiom,
    ! [C_74: nat,A_174: nat,B_134: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_174),B_134))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_74))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),A_174)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),B_134))) ) ) ).

tff(fact_1079_mult__left__mono,axiom,
    ! [C_74: real,A_174: real,B_134: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_174),B_134))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_74))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),A_174)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),B_134))) ) ) ).

tff(fact_1080_mult__left__mono,axiom,
    ! [C_74: code_code_numeral,A_174: code_code_numeral,B_134: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_174),B_134))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_74))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),A_174)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),B_134))) ) ) ).

tff(fact_1081_mult__left__mono,axiom,
    ! [C_74: quickcheck_code_int,A_174: quickcheck_code_int,B_134: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_174),B_134))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_74))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),A_174)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),B_134))) ) ) ).

tff(fact_1082_mult__left__mono,axiom,
    ! [C_74: rat,A_174: rat,B_134: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_174),B_134))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_74))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),A_174)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),B_134))) ) ) ).

tff(fact_1083_mult__right__mono,axiom,
    ! [C_73: int,A_173: int,B_133: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_173),B_133))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_73))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_173),C_73)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_133),C_73))) ) ) ).

tff(fact_1084_mult__right__mono,axiom,
    ! [C_73: nat,A_173: nat,B_133: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_173),B_133))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_73))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_173),C_73)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_133),C_73))) ) ) ).

tff(fact_1085_mult__right__mono,axiom,
    ! [C_73: real,A_173: real,B_133: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_173),B_133))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_73))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_173),C_73)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_133),C_73))) ) ) ).

tff(fact_1086_mult__right__mono,axiom,
    ! [C_73: code_code_numeral,A_173: code_code_numeral,B_133: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_173),B_133))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_73))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_173),C_73)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_133),C_73))) ) ) ).

tff(fact_1087_mult__right__mono,axiom,
    ! [C_73: quickcheck_code_int,A_173: quickcheck_code_int,B_133: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_173),B_133))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_73))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_173),C_73)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_133),C_73))) ) ) ).

tff(fact_1088_mult__right__mono,axiom,
    ! [C_73: rat,A_173: rat,B_133: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_173),B_133))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_73))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_173),C_73)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_133),C_73))) ) ) ).

tff(fact_1089_mult__nonpos__nonpos,axiom,
    ! [B_132: int,A_172: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_172),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_132),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_172),B_132))) ) ) ).

tff(fact_1090_mult__nonpos__nonpos,axiom,
    ! [B_132: real,A_172: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_172),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_132),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_172),B_132))) ) ) ).

tff(fact_1091_mult__nonpos__nonpos,axiom,
    ! [B_132: rat,A_172: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_172),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_132),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_172),B_132))) ) ) ).

tff(fact_1092_mult__nonpos__nonneg,axiom,
    ! [B_131: int,A_171: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_171),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_131))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_171),B_131)),zero_zero_int)) ) ) ).

tff(fact_1093_mult__nonpos__nonneg,axiom,
    ! [B_131: nat,A_171: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_171),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_131))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_171),B_131)),zero_zero_nat)) ) ) ).

tff(fact_1094_mult__nonpos__nonneg,axiom,
    ! [B_131: real,A_171: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_171),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_131))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_171),B_131)),zero_zero_real)) ) ) ).

tff(fact_1095_mult__nonpos__nonneg,axiom,
    ! [B_131: code_code_numeral,A_171: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_171),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_131))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_171),B_131)),zero_z126310315umeral)) ) ) ).

tff(fact_1096_mult__nonpos__nonneg,axiom,
    ! [B_131: quickcheck_code_int,A_171: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_171),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_131))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_171),B_131)),zero_z891286103de_int)) ) ) ).

tff(fact_1097_mult__nonpos__nonneg,axiom,
    ! [B_131: rat,A_171: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_171),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_131))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_171),B_131)),zero_zero_rat)) ) ) ).

tff(fact_1098_mult__nonneg__nonpos2,axiom,
    ! [B_130: int,A_170: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_170))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_130),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_130),A_170)),zero_zero_int)) ) ) ).

tff(fact_1099_mult__nonneg__nonpos2,axiom,
    ! [B_130: nat,A_170: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_170))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_130),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_130),A_170)),zero_zero_nat)) ) ) ).

tff(fact_1100_mult__nonneg__nonpos2,axiom,
    ! [B_130: real,A_170: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_170))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_130),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_130),A_170)),zero_zero_real)) ) ) ).

tff(fact_1101_mult__nonneg__nonpos2,axiom,
    ! [B_130: code_code_numeral,A_170: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_170))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_130),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_130),A_170)),zero_z126310315umeral)) ) ) ).

tff(fact_1102_mult__nonneg__nonpos2,axiom,
    ! [B_130: quickcheck_code_int,A_170: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_170))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_130),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_130),A_170)),zero_z891286103de_int)) ) ) ).

tff(fact_1103_mult__nonneg__nonpos2,axiom,
    ! [B_130: rat,A_170: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_170))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_130),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_130),A_170)),zero_zero_rat)) ) ) ).

tff(fact_1104_mult__nonneg__nonpos,axiom,
    ! [B_129: int,A_169: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_169))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_129),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_169),B_129)),zero_zero_int)) ) ) ).

tff(fact_1105_mult__nonneg__nonpos,axiom,
    ! [B_129: nat,A_169: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_169))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_129),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_169),B_129)),zero_zero_nat)) ) ) ).

tff(fact_1106_mult__nonneg__nonpos,axiom,
    ! [B_129: real,A_169: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_169))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_129),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_169),B_129)),zero_zero_real)) ) ) ).

tff(fact_1107_mult__nonneg__nonpos,axiom,
    ! [B_129: code_code_numeral,A_169: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_169))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_129),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_169),B_129)),zero_z126310315umeral)) ) ) ).

tff(fact_1108_mult__nonneg__nonpos,axiom,
    ! [B_129: quickcheck_code_int,A_169: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_169))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_129),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_169),B_129)),zero_z891286103de_int)) ) ) ).

tff(fact_1109_mult__nonneg__nonpos,axiom,
    ! [B_129: rat,A_169: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_169))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_129),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_169),B_129)),zero_zero_rat)) ) ) ).

tff(fact_1110_mult__nonneg__nonneg,axiom,
    ! [B_128: int,A_168: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_168))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_128))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_168),B_128))) ) ) ).

tff(fact_1111_mult__nonneg__nonneg,axiom,
    ! [B_128: nat,A_168: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_168))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_128))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_168),B_128))) ) ) ).

tff(fact_1112_mult__nonneg__nonneg,axiom,
    ! [B_128: real,A_168: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_168))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_128))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_168),B_128))) ) ) ).

tff(fact_1113_mult__nonneg__nonneg,axiom,
    ! [B_128: code_code_numeral,A_168: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_168))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_128))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_168),B_128))) ) ) ).

tff(fact_1114_mult__nonneg__nonneg,axiom,
    ! [B_128: quickcheck_code_int,A_168: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_168))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_128))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_168),B_128))) ) ) ).

tff(fact_1115_mult__nonneg__nonneg,axiom,
    ! [B_128: rat,A_168: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_168))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_128))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_168),B_128))) ) ) ).

tff(fact_1116_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

tff(fact_1117_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

tff(fact_1118_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

tff(fact_1119_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

tff(fact_1120_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

tff(fact_1121_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B)))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

tff(fact_1122_zero__le__square,axiom,
    ! [A_167: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_167),A_167))) ).

tff(fact_1123_zero__le__square,axiom,
    ! [A_167: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_167),A_167))) ).

tff(fact_1124_zero__le__square,axiom,
    ! [A_167: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_167),A_167))) ).

tff(fact_1125_mult__strict__left__mono__neg,axiom,
    ! [C_72: int,B_127: int,A_166: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_127),A_166))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_72),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),A_166)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),B_127))) ) ) ).

tff(fact_1126_mult__strict__left__mono__neg,axiom,
    ! [C_72: real,B_127: real,A_166: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_127),A_166))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_72),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),A_166)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),B_127))) ) ) ).

tff(fact_1127_mult__strict__left__mono__neg,axiom,
    ! [C_72: rat,B_127: rat,A_166: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_127),A_166))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_72),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),A_166)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),B_127))) ) ) ).

tff(fact_1128_mult__strict__right__mono__neg,axiom,
    ! [C_71: int,B_126: int,A_165: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_126),A_165))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_71),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_165),C_71)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_126),C_71))) ) ) ).

tff(fact_1129_mult__strict__right__mono__neg,axiom,
    ! [C_71: real,B_126: real,A_165: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_126),A_165))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_71),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_165),C_71)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_126),C_71))) ) ) ).

tff(fact_1130_mult__strict__right__mono__neg,axiom,
    ! [C_71: rat,B_126: rat,A_165: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_126),A_165))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_71),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_165),C_71)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_126),C_71))) ) ) ).

tff(fact_1131_comm__mult__strict__left__mono,axiom,
    ! [C_70: int,A_164: int,B_125: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_164),B_125))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_70))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),A_164)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),B_125))) ) ) ).

tff(fact_1132_comm__mult__strict__left__mono,axiom,
    ! [C_70: nat,A_164: nat,B_125: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_164),B_125))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_70))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),A_164)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),B_125))) ) ) ).

tff(fact_1133_comm__mult__strict__left__mono,axiom,
    ! [C_70: real,A_164: real,B_125: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_164),B_125))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_70))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),A_164)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),B_125))) ) ) ).

tff(fact_1134_comm__mult__strict__left__mono,axiom,
    ! [C_70: code_code_numeral,A_164: code_code_numeral,B_125: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_164),B_125))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_70))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),A_164)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),B_125))) ) ) ).

tff(fact_1135_comm__mult__strict__left__mono,axiom,
    ! [C_70: quickcheck_code_int,A_164: quickcheck_code_int,B_125: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_164),B_125))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_70))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),A_164)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),B_125))) ) ) ).

tff(fact_1136_comm__mult__strict__left__mono,axiom,
    ! [C_70: rat,A_164: rat,B_125: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_164),B_125))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_70))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),A_164)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),B_125))) ) ) ).

tff(fact_1137_mult__strict__left__mono,axiom,
    ! [C_69: int,A_163: int,B_124: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_163),B_124))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_69))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),A_163)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),B_124))) ) ) ).

tff(fact_1138_mult__strict__left__mono,axiom,
    ! [C_69: nat,A_163: nat,B_124: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_163),B_124))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_69))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),A_163)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),B_124))) ) ) ).

tff(fact_1139_mult__strict__left__mono,axiom,
    ! [C_69: real,A_163: real,B_124: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_163),B_124))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_69))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),A_163)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),B_124))) ) ) ).

tff(fact_1140_mult__strict__left__mono,axiom,
    ! [C_69: code_code_numeral,A_163: code_code_numeral,B_124: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_163),B_124))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_69))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),A_163)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),B_124))) ) ) ).

tff(fact_1141_mult__strict__left__mono,axiom,
    ! [C_69: quickcheck_code_int,A_163: quickcheck_code_int,B_124: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_163),B_124))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_69))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),A_163)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),B_124))) ) ) ).

tff(fact_1142_mult__strict__left__mono,axiom,
    ! [C_69: rat,A_163: rat,B_124: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_163),B_124))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_69))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),A_163)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),B_124))) ) ) ).

tff(fact_1143_mult__strict__right__mono,axiom,
    ! [C_68: int,A_162: int,B_123: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_162),B_123))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_68))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_162),C_68)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_123),C_68))) ) ) ).

tff(fact_1144_mult__strict__right__mono,axiom,
    ! [C_68: nat,A_162: nat,B_123: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_162),B_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_68))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_162),C_68)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_123),C_68))) ) ) ).

tff(fact_1145_mult__strict__right__mono,axiom,
    ! [C_68: real,A_162: real,B_123: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_162),B_123))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_68))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_162),C_68)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_123),C_68))) ) ) ).

tff(fact_1146_mult__strict__right__mono,axiom,
    ! [C_68: code_code_numeral,A_162: code_code_numeral,B_123: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_162),B_123))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_68))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_162),C_68)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_123),C_68))) ) ) ).

tff(fact_1147_mult__strict__right__mono,axiom,
    ! [C_68: quickcheck_code_int,A_162: quickcheck_code_int,B_123: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_162),B_123))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_68))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_162),C_68)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_123),C_68))) ) ) ).

tff(fact_1148_mult__strict__right__mono,axiom,
    ! [C_68: rat,A_162: rat,B_123: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_162),B_123))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_68))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_162),C_68)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_123),C_68))) ) ) ).

tff(fact_1149_mult__neg__neg,axiom,
    ! [B_122: int,A_161: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_161),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_122),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_161),B_122))) ) ) ).

tff(fact_1150_mult__neg__neg,axiom,
    ! [B_122: real,A_161: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_161),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_122),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_161),B_122))) ) ) ).

tff(fact_1151_mult__neg__neg,axiom,
    ! [B_122: rat,A_161: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_161),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_122),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_161),B_122))) ) ) ).

tff(fact_1152_mult__neg__pos,axiom,
    ! [B_121: int,A_160: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_160),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_121))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_160),B_121)),zero_zero_int)) ) ) ).

tff(fact_1153_mult__neg__pos,axiom,
    ! [B_121: nat,A_160: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_160),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_121))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_160),B_121)),zero_zero_nat)) ) ) ).

tff(fact_1154_mult__neg__pos,axiom,
    ! [B_121: real,A_160: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_160),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_121))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_160),B_121)),zero_zero_real)) ) ) ).

tff(fact_1155_mult__neg__pos,axiom,
    ! [B_121: code_code_numeral,A_160: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_160),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_121))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_160),B_121)),zero_z126310315umeral)) ) ) ).

tff(fact_1156_mult__neg__pos,axiom,
    ! [B_121: quickcheck_code_int,A_160: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_160),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_121))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_160),B_121)),zero_z891286103de_int)) ) ) ).

tff(fact_1157_mult__neg__pos,axiom,
    ! [B_121: rat,A_160: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_160),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_121))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_160),B_121)),zero_zero_rat)) ) ) ).

tff(fact_1158_mult__less__cancel__left__neg,axiom,
    ! [A: int,B: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ).

tff(fact_1159_mult__less__cancel__left__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) ) ) ).

tff(fact_1160_mult__less__cancel__left__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) ) ) ).

tff(fact_1161_zero__less__mult__pos2,axiom,
    ! [B_120: int,A_159: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_120),A_159)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_159))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_120)) ) ) ).

tff(fact_1162_zero__less__mult__pos2,axiom,
    ! [B_120: nat,A_159: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_120),A_159)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_159))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_120)) ) ) ).

tff(fact_1163_zero__less__mult__pos2,axiom,
    ! [B_120: real,A_159: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_120),A_159)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_159))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_120)) ) ) ).

tff(fact_1164_zero__less__mult__pos2,axiom,
    ! [B_120: code_code_numeral,A_159: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_120),A_159)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_159))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_120)) ) ) ).

tff(fact_1165_zero__less__mult__pos2,axiom,
    ! [B_120: quickcheck_code_int,A_159: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_120),A_159)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_159))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_120)) ) ) ).

tff(fact_1166_zero__less__mult__pos2,axiom,
    ! [B_120: rat,A_159: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_120),A_159)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_159))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_120)) ) ) ).

tff(fact_1167_zero__less__mult__pos,axiom,
    ! [A_158: int,B_119: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_158),B_119)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_158))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_119)) ) ) ).

tff(fact_1168_zero__less__mult__pos,axiom,
    ! [A_158: nat,B_119: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_158),B_119)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_158))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_119)) ) ) ).

tff(fact_1169_zero__less__mult__pos,axiom,
    ! [A_158: real,B_119: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_158),B_119)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_158))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_119)) ) ) ).

tff(fact_1170_zero__less__mult__pos,axiom,
    ! [A_158: code_code_numeral,B_119: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_158),B_119)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_158))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_119)) ) ) ).

tff(fact_1171_zero__less__mult__pos,axiom,
    ! [A_158: quickcheck_code_int,B_119: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_158),B_119)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_158))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_119)) ) ) ).

tff(fact_1172_zero__less__mult__pos,axiom,
    ! [A_158: rat,B_119: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_158),B_119)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_158))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_119)) ) ) ).

tff(fact_1173_mult__pos__neg2,axiom,
    ! [B_118: int,A_157: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_157))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_118),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_118),A_157)),zero_zero_int)) ) ) ).

tff(fact_1174_mult__pos__neg2,axiom,
    ! [B_118: nat,A_157: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_157))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_118),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_118),A_157)),zero_zero_nat)) ) ) ).

tff(fact_1175_mult__pos__neg2,axiom,
    ! [B_118: real,A_157: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_157))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_118),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_118),A_157)),zero_zero_real)) ) ) ).

tff(fact_1176_mult__pos__neg2,axiom,
    ! [B_118: code_code_numeral,A_157: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_157))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_118),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_118),A_157)),zero_z126310315umeral)) ) ) ).

tff(fact_1177_mult__pos__neg2,axiom,
    ! [B_118: quickcheck_code_int,A_157: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_157))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_118),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_118),A_157)),zero_z891286103de_int)) ) ) ).

tff(fact_1178_mult__pos__neg2,axiom,
    ! [B_118: rat,A_157: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_157))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_118),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_118),A_157)),zero_zero_rat)) ) ) ).

tff(fact_1179_mult__pos__neg,axiom,
    ! [B_117: int,A_156: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_156))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_117),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_156),B_117)),zero_zero_int)) ) ) ).

tff(fact_1180_mult__pos__neg,axiom,
    ! [B_117: nat,A_156: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_156))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_117),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_156),B_117)),zero_zero_nat)) ) ) ).

tff(fact_1181_mult__pos__neg,axiom,
    ! [B_117: real,A_156: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_156))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_117),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_156),B_117)),zero_zero_real)) ) ) ).

tff(fact_1182_mult__pos__neg,axiom,
    ! [B_117: code_code_numeral,A_156: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_156))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_117),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_156),B_117)),zero_z126310315umeral)) ) ) ).

tff(fact_1183_mult__pos__neg,axiom,
    ! [B_117: quickcheck_code_int,A_156: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_156))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_117),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_156),B_117)),zero_z891286103de_int)) ) ) ).

tff(fact_1184_mult__pos__neg,axiom,
    ! [B_117: rat,A_156: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_156))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_117),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_156),B_117)),zero_zero_rat)) ) ) ).

tff(fact_1185_mult__pos__pos,axiom,
    ! [B_116: int,A_155: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_155))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_155),B_116))) ) ) ).

tff(fact_1186_mult__pos__pos,axiom,
    ! [B_116: nat,A_155: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_155))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_155),B_116))) ) ) ).

tff(fact_1187_mult__pos__pos,axiom,
    ! [B_116: real,A_155: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_155))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_116))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_155),B_116))) ) ) ).

tff(fact_1188_mult__pos__pos,axiom,
    ! [B_116: code_code_numeral,A_155: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_155))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_155),B_116))) ) ) ).

tff(fact_1189_mult__pos__pos,axiom,
    ! [B_116: quickcheck_code_int,A_155: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_155))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_155),B_116))) ) ) ).

tff(fact_1190_mult__pos__pos,axiom,
    ! [B_116: rat,A_155: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_155))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_116))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_155),B_116))) ) ) ).

tff(fact_1191_mult__less__cancel__left__pos,axiom,
    ! [A: int,B: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ) ).

tff(fact_1192_mult__less__cancel__left__pos,axiom,
    ! [A: real,B: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ) ).

tff(fact_1193_mult__less__cancel__left__pos,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) ) ) ).

tff(fact_1194_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ) ).

tff(fact_1195_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) ) ) ) ).

tff(fact_1196_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) ) ) ) ).

tff(fact_1197_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ) ).

tff(fact_1198_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) )
        | ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) ) ) ) ).

tff(fact_1199_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) )
        | ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
          & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) ) ) ) ).

tff(fact_1200_not__square__less__zero,axiom,
    ! [A_154: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_154),A_154)),zero_zero_int)) ).

tff(fact_1201_not__square__less__zero,axiom,
    ! [A_154: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_154),A_154)),zero_zero_real)) ).

tff(fact_1202_not__square__less__zero,axiom,
    ! [A_154: rat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_154),A_154)),zero_zero_rat)) ).

tff(fact_1203_pos__add__strict,axiom,
    ! [B_115: code_code_numeral,C_67: code_code_numeral,A_153: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_153))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_115),C_67))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_115),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_153),C_67))) ) ) ).

tff(fact_1204_pos__add__strict,axiom,
    ! [B_115: int,C_67: int,A_153: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_153))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),C_67))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_153),C_67))) ) ) ).

tff(fact_1205_pos__add__strict,axiom,
    ! [B_115: nat,C_67: nat,A_153: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_153))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_115),C_67))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_115),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_153),C_67))) ) ) ).

tff(fact_1206_pos__add__strict,axiom,
    ! [B_115: real,C_67: real,A_153: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_153))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),C_67))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_153),C_67))) ) ) ).

tff(fact_1207_pos__add__strict,axiom,
    ! [B_115: quickcheck_code_int,C_67: quickcheck_code_int,A_153: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_153))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_115),C_67))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_115),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_153),C_67))) ) ) ).

tff(fact_1208_pos__add__strict,axiom,
    ! [B_115: rat,C_67: rat,A_153: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_153))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),C_67))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_153),C_67))) ) ) ).

tff(fact_1209_zero__le__one,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

tff(fact_1210_zero__le__one,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

tff(fact_1211_zero__le__one,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

tff(fact_1212_zero__le__one,axiom,
    hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),one_on1645066479umeral)) ).

tff(fact_1213_zero__le__one,axiom,
    hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),one_on1684967323de_int)) ).

tff(fact_1214_zero__le__one,axiom,
    hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

tff(fact_1215_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

tff(fact_1216_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

tff(fact_1217_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

tff(fact_1218_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),zero_z126310315umeral)) ).

tff(fact_1219_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),zero_z891286103de_int)) ).

tff(fact_1220_not__one__le__zero,axiom,
    ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

tff(fact_1221_zero__less__one,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

tff(fact_1222_zero__less__one,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

tff(fact_1223_zero__less__one,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

tff(fact_1224_zero__less__one,axiom,
    hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),one_on1645066479umeral)) ).

tff(fact_1225_zero__less__one,axiom,
    hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),one_on1684967323de_int)) ).

tff(fact_1226_zero__less__one,axiom,
    hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

tff(fact_1227_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).

tff(fact_1228_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

tff(fact_1229_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

tff(fact_1230_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),zero_z126310315umeral)) ).

tff(fact_1231_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),zero_z891286103de_int)) ).

tff(fact_1232_not__one__less__zero,axiom,
    ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

tff(fact_1233_power__mono,axiom,
    ! [N_55: nat,A_152: int,B_114: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_152),B_114))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_152))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_152),N_55)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_114),N_55))) ) ) ).

tff(fact_1234_power__mono,axiom,
    ! [N_55: nat,A_152: nat,B_114: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_152),B_114))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_152))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_152),N_55)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_114),N_55))) ) ) ).

tff(fact_1235_power__mono,axiom,
    ! [N_55: nat,A_152: real,B_114: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_152),B_114))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_152))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_152),N_55)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_114),N_55))) ) ) ).

tff(fact_1236_power__mono,axiom,
    ! [N_55: nat,A_152: code_code_numeral,B_114: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_152),B_114))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_152))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_152),N_55)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_114),N_55))) ) ) ).

tff(fact_1237_power__mono,axiom,
    ! [N_55: nat,A_152: quickcheck_code_int,B_114: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_152),B_114))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_152))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_152),N_55)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_114),N_55))) ) ) ).

tff(fact_1238_power__mono,axiom,
    ! [N_55: nat,A_152: rat,B_114: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_152),B_114))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_152))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_152),N_55)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_114),N_55))) ) ) ).

tff(fact_1239_zero__le__power,axiom,
    ! [N_54: nat,A_151: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_151))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_151),N_54))) ) ).

tff(fact_1240_zero__le__power,axiom,
    ! [N_54: nat,A_151: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_151))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_151),N_54))) ) ).

tff(fact_1241_zero__le__power,axiom,
    ! [N_54: nat,A_151: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_151))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_151),N_54))) ) ).

tff(fact_1242_zero__le__power,axiom,
    ! [N_54: nat,A_151: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_151))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_151),N_54))) ) ).

tff(fact_1243_zero__le__power,axiom,
    ! [N_54: nat,A_151: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_151))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_151),N_54))) ) ).

tff(fact_1244_zero__le__power,axiom,
    ! [N_54: nat,A_151: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_151))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_151),N_54))) ) ).

tff(fact_1245_less__1__mult,axiom,
    ! [N_53: int,M_21: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_21))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_53))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_21),N_53))) ) ) ).

tff(fact_1246_less__1__mult,axiom,
    ! [N_53: nat,M_21: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_21))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_53))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_21),N_53))) ) ) ).

tff(fact_1247_less__1__mult,axiom,
    ! [N_53: real,M_21: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_21))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_53))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_21),N_53))) ) ) ).

tff(fact_1248_less__1__mult,axiom,
    ! [N_53: code_code_numeral,M_21: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_21))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_53))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_21),N_53))) ) ) ).

tff(fact_1249_less__1__mult,axiom,
    ! [N_53: quickcheck_code_int,M_21: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_21))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_53))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_21),N_53))) ) ) ).

tff(fact_1250_less__1__mult,axiom,
    ! [N_53: rat,M_21: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_21))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_53))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_21),N_53))) ) ) ).

tff(fact_1251_zero__less__power,axiom,
    ! [N_52: nat,A_150: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_150))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_150),N_52))) ) ).

tff(fact_1252_zero__less__power,axiom,
    ! [N_52: nat,A_150: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_150))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_150),N_52))) ) ).

tff(fact_1253_zero__less__power,axiom,
    ! [N_52: nat,A_150: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_150))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_150),N_52))) ) ).

tff(fact_1254_zero__less__power,axiom,
    ! [N_52: nat,A_150: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_150))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_150),N_52))) ) ).

tff(fact_1255_zero__less__power,axiom,
    ! [N_52: nat,A_150: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_150))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_150),N_52))) ) ).

tff(fact_1256_zero__less__power,axiom,
    ! [N_52: nat,A_150: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_150))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_150),N_52))) ) ).

tff(fact_1257_less__add__one,axiom,
    ! [A_149: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_149),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_149),one_one_int))) ).

tff(fact_1258_less__add__one,axiom,
    ! [A_149: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_149),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_149),one_one_nat))) ).

tff(fact_1259_less__add__one,axiom,
    ! [A_149: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_149),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_149),one_one_real))) ).

tff(fact_1260_less__add__one,axiom,
    ! [A_149: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_149),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_149),one_on1645066479umeral))) ).

tff(fact_1261_less__add__one,axiom,
    ! [A_149: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_149),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_149),one_on1684967323de_int))) ).

tff(fact_1262_less__add__one,axiom,
    ! [A_149: rat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_149),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_149),one_one_rat))) ).

tff(fact_1263_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
    <=> ( ( C = zero_zero_real )
        | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) ) ) ).

tff(fact_1264_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),A)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),B)))
    <=> ( ( C = zero_zero_complex )
        | hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) ) ) ).

tff(fact_1265_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
    <=> ( ( C = zero_zero_rat )
        | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)) ) ) ).

tff(fact_1266_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( C = zero_zero_int )
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ).

tff(fact_1267_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)))
    <=> ( ( C = zero_zero_real )
        | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) ) ) ).

tff(fact_1268_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C)))
    <=> ( ( C = zero_zero_complex )
        | hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) ) ) ).

tff(fact_1269_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)))
    <=> ( ( C = zero_zero_rat )
        | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)) ) ) ).

tff(fact_1270_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( C = zero_zero_int )
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ).

tff(fact_1271_power__increasing,axiom,
    ! [A_148: int,N_51: nat,N_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_148))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_148),N_51)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_148),N_50))) ) ) ).

tff(fact_1272_power__increasing,axiom,
    ! [A_148: nat,N_51: nat,N_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_148))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_148),N_51)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_148),N_50))) ) ) ).

tff(fact_1273_power__increasing,axiom,
    ! [A_148: real,N_51: nat,N_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_148))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_148),N_51)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_148),N_50))) ) ) ).

tff(fact_1274_power__increasing,axiom,
    ! [A_148: code_code_numeral,N_51: nat,N_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_148))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_148),N_51)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_148),N_50))) ) ) ).

tff(fact_1275_power__increasing,axiom,
    ! [A_148: quickcheck_code_int,N_51: nat,N_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_148))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_148),N_51)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_148),N_50))) ) ) ).

tff(fact_1276_power__increasing,axiom,
    ! [A_148: rat,N_51: nat,N_50: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_148))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_148),N_51)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_148),N_50))) ) ) ).

tff(fact_1277_one__le__power,axiom,
    ! [N_49: nat,A_147: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_147))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_147),N_49))) ) ).

tff(fact_1278_one__le__power,axiom,
    ! [N_49: nat,A_147: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_147))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_147),N_49))) ) ).

tff(fact_1279_one__le__power,axiom,
    ! [N_49: nat,A_147: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_147))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_147),N_49))) ) ).

tff(fact_1280_one__le__power,axiom,
    ! [N_49: nat,A_147: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_147))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_147),N_49))) ) ).

tff(fact_1281_one__le__power,axiom,
    ! [N_49: nat,A_147: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_147))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_147),N_49))) ) ).

tff(fact_1282_one__le__power,axiom,
    ! [N_49: nat,A_147: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_147))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_147),N_49))) ) ).

tff(fact_1283_power__inject__exp,axiom,
    ! [Ma: nat,N_1: nat,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
     => ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),Ma) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1284_power__inject__exp,axiom,
    ! [Ma: nat,N_1: nat,A: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1285_power__inject__exp,axiom,
    ! [Ma: nat,N_1: nat,A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A))
     => ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),Ma) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1286_power__inject__exp,axiom,
    ! [Ma: nat,N_1: nat,A: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A))
     => ( ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),Ma) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1287_power__inject__exp,axiom,
    ! [Ma: nat,N_1: nat,A: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A))
     => ( ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),Ma) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1288_power__inject__exp,axiom,
    ! [Ma: nat,N_1: nat,A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A))
     => ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),Ma) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1289_power__eq__0__iff,axiom,
    ! [A: int,N_1: nat] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N_1) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1290_power__eq__0__iff,axiom,
    ! [A: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N_1) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1291_power__eq__0__iff,axiom,
    ! [A: real,N_1: nat] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N_1) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1292_power__eq__0__iff,axiom,
    ! [A: code_code_numeral,N_1: nat] :
      ( ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N_1) = zero_z126310315umeral )
    <=> ( ( A = zero_z126310315umeral )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1293_power__eq__0__iff,axiom,
    ! [A: complex,N_1: nat] :
      ( ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),N_1) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1294_power__eq__0__iff,axiom,
    ! [A: quickcheck_code_int,N_1: nat] :
      ( ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N_1) = zero_z891286103de_int )
    <=> ( ( A = zero_z891286103de_int )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1295_power__eq__0__iff,axiom,
    ! [A: rat,N_1: nat] :
      ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N_1) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_1296_power__0,axiom,
    ! [A_146: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_146),zero_zero_nat) = one_one_int ).

tff(fact_1297_power__0,axiom,
    ! [A_146: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_146),zero_zero_nat) = one_one_nat ).

tff(fact_1298_power__0,axiom,
    ! [A_146: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_146),zero_zero_nat) = one_one_real ).

tff(fact_1299_power__0,axiom,
    ! [A_146: code_code_numeral] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_146),zero_zero_nat) = one_on1645066479umeral ).

tff(fact_1300_power__0,axiom,
    ! [A_146: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_146),zero_zero_nat) = one_one_complex ).

tff(fact_1301_power__0,axiom,
    ! [A_146: quickcheck_code_int] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_146),zero_zero_nat) = one_on1684967323de_int ).

tff(fact_1302_power__0,axiom,
    ! [A_146: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_146),zero_zero_nat) = one_one_rat ).

tff(fact_1303_power__add,axiom,
    ! [A_145: code_code_numeral,M_20: nat,N_48: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_145),M_20)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_145),N_48)) ).

tff(fact_1304_power__add,axiom,
    ! [A_145: quickcheck_code_int,M_20: nat,N_48: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_145),M_20)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_145),N_48)) ).

tff(fact_1305_power__add,axiom,
    ! [A_145: rat,M_20: nat,N_48: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_145),M_20)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_145),N_48)) ).

tff(fact_1306_power__add,axiom,
    ! [A_145: int,M_20: nat,N_48: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_145),M_20)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_145),N_48)) ).

tff(fact_1307_power__add,axiom,
    ! [A_145: nat,M_20: nat,N_48: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_145),M_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_145),N_48)) ).

tff(fact_1308_power__add,axiom,
    ! [A_145: real,M_20: nat,N_48: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_145),M_20)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_145),N_48)) ).

tff(fact_1309_power__add,axiom,
    ! [A_145: complex,M_20: nat,N_48: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_145),M_20)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_145),N_48)) ).

tff(fact_1310_zcong__not,axiom,
    ! [B_2: int,M: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3))
           => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) ) ) ) ) ).

tff(fact_1311_zcong__not__zero,axiom,
    ! [M: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
       => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M)) ) ) ).

tff(fact_1312_zcong__less__eq,axiom,
    ! [M: int,Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M))
               => ( X = Y ) ) ) ) ) ) ) ).

tff(fact_1313_zdvd__bounds,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int))
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M)) ) ) ).

tff(fact_1314_zcong__iff__lin,axiom,
    ! [A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
    <=> ? [K: int] : B = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),K)) ) ).

tff(fact_1315_zcong__eq__zdvd__prop,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),X_1)) ) ).

tff(fact_1316_zcong__zero__equiv__div,axiom,
    ! [A: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),Ma))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A)) ) ).

tff(fact_1317_zprime__zdvd__zmult__better,axiom,
    ! [M: int,N: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N)) ) ) ) ).

tff(fact_1318_mult__left__le__imp__le,axiom,
    ! [C_66: int,A_144: int,B_113: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_66),A_144)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_66),B_113)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_66))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_144),B_113)) ) ) ).

tff(fact_1319_mult__left__le__imp__le,axiom,
    ! [C_66: nat,A_144: nat,B_113: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_66),A_144)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_66),B_113)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_66))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_144),B_113)) ) ) ).

tff(fact_1320_mult__left__le__imp__le,axiom,
    ! [C_66: real,A_144: real,B_113: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_66),A_144)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_66),B_113)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_66))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_144),B_113)) ) ) ).

tff(fact_1321_mult__left__le__imp__le,axiom,
    ! [C_66: code_code_numeral,A_144: code_code_numeral,B_113: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_66),A_144)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_66),B_113)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_66))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_144),B_113)) ) ) ).

tff(fact_1322_mult__left__le__imp__le,axiom,
    ! [C_66: quickcheck_code_int,A_144: quickcheck_code_int,B_113: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_66),A_144)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_66),B_113)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_66))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_144),B_113)) ) ) ).

tff(fact_1323_mult__left__le__imp__le,axiom,
    ! [C_66: rat,A_144: rat,B_113: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_66),A_144)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_66),B_113)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_66))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_144),B_113)) ) ) ).

tff(fact_1324_mult__right__le__imp__le,axiom,
    ! [A_143: int,C_65: int,B_112: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_143),C_65)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_112),C_65)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_65))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_143),B_112)) ) ) ).

tff(fact_1325_mult__right__le__imp__le,axiom,
    ! [A_143: nat,C_65: nat,B_112: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_143),C_65)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_112),C_65)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_65))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_143),B_112)) ) ) ).

tff(fact_1326_mult__right__le__imp__le,axiom,
    ! [A_143: real,C_65: real,B_112: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_143),C_65)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_112),C_65)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_65))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_143),B_112)) ) ) ).

tff(fact_1327_mult__right__le__imp__le,axiom,
    ! [A_143: code_code_numeral,C_65: code_code_numeral,B_112: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_143),C_65)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_112),C_65)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_65))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_143),B_112)) ) ) ).

tff(fact_1328_mult__right__le__imp__le,axiom,
    ! [A_143: quickcheck_code_int,C_65: quickcheck_code_int,B_112: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_143),C_65)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_112),C_65)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_65))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_143),B_112)) ) ) ).

tff(fact_1329_mult__right__le__imp__le,axiom,
    ! [A_143: rat,C_65: rat,B_112: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_143),C_65)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_112),C_65)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_65))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_143),B_112)) ) ) ).

tff(fact_1330_mult__less__imp__less__left,axiom,
    ! [C_64: int,A_142: int,B_111: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_64),A_142)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_64),B_111)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_64))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),B_111)) ) ) ).

tff(fact_1331_mult__less__imp__less__left,axiom,
    ! [C_64: nat,A_142: nat,B_111: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_64),A_142)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_64),B_111)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_64))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_142),B_111)) ) ) ).

tff(fact_1332_mult__less__imp__less__left,axiom,
    ! [C_64: real,A_142: real,B_111: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_64),A_142)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_64),B_111)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_64))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),B_111)) ) ) ).

tff(fact_1333_mult__less__imp__less__left,axiom,
    ! [C_64: code_code_numeral,A_142: code_code_numeral,B_111: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_64),A_142)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_64),B_111)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_64))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_142),B_111)) ) ) ).

tff(fact_1334_mult__less__imp__less__left,axiom,
    ! [C_64: quickcheck_code_int,A_142: quickcheck_code_int,B_111: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_64),A_142)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_64),B_111)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_64))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_142),B_111)) ) ) ).

tff(fact_1335_mult__less__imp__less__left,axiom,
    ! [C_64: rat,A_142: rat,B_111: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_64),A_142)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_64),B_111)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_64))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),B_111)) ) ) ).

tff(fact_1336_mult__left__less__imp__less,axiom,
    ! [C_63: int,A_141: int,B_110: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_63),A_141)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_63),B_110)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_63))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_141),B_110)) ) ) ).

tff(fact_1337_mult__left__less__imp__less,axiom,
    ! [C_63: nat,A_141: nat,B_110: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_63),A_141)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_63),B_110)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_63))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_141),B_110)) ) ) ).

tff(fact_1338_mult__left__less__imp__less,axiom,
    ! [C_63: real,A_141: real,B_110: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_63),A_141)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_63),B_110)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_63))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_141),B_110)) ) ) ).

tff(fact_1339_mult__left__less__imp__less,axiom,
    ! [C_63: code_code_numeral,A_141: code_code_numeral,B_110: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_63),A_141)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_63),B_110)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_63))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_141),B_110)) ) ) ).

tff(fact_1340_mult__left__less__imp__less,axiom,
    ! [C_63: quickcheck_code_int,A_141: quickcheck_code_int,B_110: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_63),A_141)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_63),B_110)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_63))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_141),B_110)) ) ) ).

tff(fact_1341_mult__left__less__imp__less,axiom,
    ! [C_63: rat,A_141: rat,B_110: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_63),A_141)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_63),B_110)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_63))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_141),B_110)) ) ) ).

tff(fact_1342_mult__less__imp__less__right,axiom,
    ! [A_140: int,C_62: int,B_109: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_140),C_62)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_109),C_62)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_62))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_140),B_109)) ) ) ).

tff(fact_1343_mult__less__imp__less__right,axiom,
    ! [A_140: nat,C_62: nat,B_109: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_140),C_62)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_109),C_62)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_62))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_140),B_109)) ) ) ).

tff(fact_1344_mult__less__imp__less__right,axiom,
    ! [A_140: real,C_62: real,B_109: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_140),C_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_109),C_62)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_62))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_140),B_109)) ) ) ).

tff(fact_1345_mult__less__imp__less__right,axiom,
    ! [A_140: code_code_numeral,C_62: code_code_numeral,B_109: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_140),C_62)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_109),C_62)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_62))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_140),B_109)) ) ) ).

tff(fact_1346_mult__less__imp__less__right,axiom,
    ! [A_140: quickcheck_code_int,C_62: quickcheck_code_int,B_109: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_140),C_62)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_109),C_62)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_62))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_140),B_109)) ) ) ).

tff(fact_1347_mult__less__imp__less__right,axiom,
    ! [A_140: rat,C_62: rat,B_109: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_140),C_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_109),C_62)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_62))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_140),B_109)) ) ) ).

tff(fact_1348_mult__right__less__imp__less,axiom,
    ! [A_139: int,C_61: int,B_108: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),C_61)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_108),C_61)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_61))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_139),B_108)) ) ) ).

tff(fact_1349_mult__right__less__imp__less,axiom,
    ! [A_139: nat,C_61: nat,B_108: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),C_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_108),C_61)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_61))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_139),B_108)) ) ) ).

tff(fact_1350_mult__right__less__imp__less,axiom,
    ! [A_139: real,C_61: real,B_108: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),C_61)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_108),C_61)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_61))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_139),B_108)) ) ) ).

tff(fact_1351_mult__right__less__imp__less,axiom,
    ! [A_139: code_code_numeral,C_61: code_code_numeral,B_108: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_139),C_61)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_108),C_61)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_61))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_139),B_108)) ) ) ).

tff(fact_1352_mult__right__less__imp__less,axiom,
    ! [A_139: quickcheck_code_int,C_61: quickcheck_code_int,B_108: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),C_61)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_108),C_61)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_61))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_139),B_108)) ) ) ).

tff(fact_1353_mult__right__less__imp__less,axiom,
    ! [A_139: rat,C_61: rat,B_108: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_139),C_61)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_108),C_61)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_61))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_139),B_108)) ) ) ).

tff(fact_1354_mult__le__less__imp__less,axiom,
    ! [C_60: int,D_13: int,A_138: int,B_107: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_138),B_107))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_60),D_13))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_138))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_60))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_138),C_60)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_107),D_13))) ) ) ) ) ).

tff(fact_1355_mult__le__less__imp__less,axiom,
    ! [C_60: nat,D_13: nat,A_138: nat,B_107: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_138),B_107))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_60),D_13))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_138))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_60))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_138),C_60)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_107),D_13))) ) ) ) ) ).

tff(fact_1356_mult__le__less__imp__less,axiom,
    ! [C_60: real,D_13: real,A_138: real,B_107: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_138),B_107))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_60),D_13))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_138))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_60))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_138),C_60)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_107),D_13))) ) ) ) ) ).

tff(fact_1357_mult__le__less__imp__less,axiom,
    ! [C_60: code_code_numeral,D_13: code_code_numeral,A_138: code_code_numeral,B_107: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_138),B_107))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_60),D_13))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_138))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_60))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_138),C_60)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_107),D_13))) ) ) ) ) ).

tff(fact_1358_mult__le__less__imp__less,axiom,
    ! [C_60: quickcheck_code_int,D_13: quickcheck_code_int,A_138: quickcheck_code_int,B_107: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_138),B_107))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_60),D_13))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_138))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_60))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_138),C_60)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_107),D_13))) ) ) ) ) ).

tff(fact_1359_mult__le__less__imp__less,axiom,
    ! [C_60: rat,D_13: rat,A_138: rat,B_107: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_138),B_107))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_60),D_13))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_138))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_60))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_138),C_60)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_107),D_13))) ) ) ) ) ).

tff(fact_1360_mult__less__le__imp__less,axiom,
    ! [C_59: int,D_12: int,A_137: int,B_106: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_137),B_106))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_59),D_12))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_137))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_59))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_137),C_59)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_106),D_12))) ) ) ) ) ).

tff(fact_1361_mult__less__le__imp__less,axiom,
    ! [C_59: nat,D_12: nat,A_137: nat,B_106: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_137),B_106))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_59),D_12))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_137))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_59))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_137),C_59)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_106),D_12))) ) ) ) ) ).

tff(fact_1362_mult__less__le__imp__less,axiom,
    ! [C_59: real,D_12: real,A_137: real,B_106: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_137),B_106))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_59),D_12))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_137))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_59))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_137),C_59)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_106),D_12))) ) ) ) ) ).

tff(fact_1363_mult__less__le__imp__less,axiom,
    ! [C_59: code_code_numeral,D_12: code_code_numeral,A_137: code_code_numeral,B_106: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_137),B_106))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_59),D_12))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_137))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_59))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_137),C_59)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_106),D_12))) ) ) ) ) ).

tff(fact_1364_mult__less__le__imp__less,axiom,
    ! [C_59: quickcheck_code_int,D_12: quickcheck_code_int,A_137: quickcheck_code_int,B_106: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_137),B_106))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_59),D_12))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_137))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_59))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_137),C_59)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_106),D_12))) ) ) ) ) ).

tff(fact_1365_mult__less__le__imp__less,axiom,
    ! [C_59: rat,D_12: rat,A_137: rat,B_106: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_137),B_106))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_59),D_12))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_137))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_59))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),C_59)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_106),D_12))) ) ) ) ) ).

tff(fact_1366_mult__strict__mono_H,axiom,
    ! [C_58: int,D_11: int,A_136: int,B_105: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_136),B_105))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_58),D_11))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_136))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_58))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_136),C_58)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_105),D_11))) ) ) ) ) ).

tff(fact_1367_mult__strict__mono_H,axiom,
    ! [C_58: nat,D_11: nat,A_136: nat,B_105: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_136),B_105))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_58),D_11))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_136))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_58))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),C_58)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_105),D_11))) ) ) ) ) ).

tff(fact_1368_mult__strict__mono_H,axiom,
    ! [C_58: real,D_11: real,A_136: real,B_105: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_136),B_105))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_58),D_11))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_136))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_58))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_136),C_58)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_105),D_11))) ) ) ) ) ).

tff(fact_1369_mult__strict__mono_H,axiom,
    ! [C_58: code_code_numeral,D_11: code_code_numeral,A_136: code_code_numeral,B_105: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_136),B_105))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_58),D_11))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_136))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_58))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_136),C_58)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_105),D_11))) ) ) ) ) ).

tff(fact_1370_mult__strict__mono_H,axiom,
    ! [C_58: quickcheck_code_int,D_11: quickcheck_code_int,A_136: quickcheck_code_int,B_105: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_136),B_105))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_58),D_11))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_136))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_58))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),C_58)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_105),D_11))) ) ) ) ) ).

tff(fact_1371_mult__strict__mono_H,axiom,
    ! [C_58: rat,D_11: rat,A_136: rat,B_105: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_136),B_105))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_58),D_11))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_136))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_58))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_136),C_58)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_105),D_11))) ) ) ) ) ).

tff(fact_1372_mult__strict__mono,axiom,
    ! [C_57: int,D_10: int,A_135: int,B_104: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_135),B_104))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_57),D_10))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_104))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_57))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_135),C_57)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_104),D_10))) ) ) ) ) ).

tff(fact_1373_mult__strict__mono,axiom,
    ! [C_57: nat,D_10: nat,A_135: nat,B_104: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_135),B_104))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_57),D_10))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_104))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_57))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_135),C_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_104),D_10))) ) ) ) ) ).

tff(fact_1374_mult__strict__mono,axiom,
    ! [C_57: real,D_10: real,A_135: real,B_104: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_135),B_104))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_57),D_10))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_104))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_57))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_135),C_57)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_104),D_10))) ) ) ) ) ).

tff(fact_1375_mult__strict__mono,axiom,
    ! [C_57: code_code_numeral,D_10: code_code_numeral,A_135: code_code_numeral,B_104: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_135),B_104))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_57),D_10))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_104))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_57))
           => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_135),C_57)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_104),D_10))) ) ) ) ) ).

tff(fact_1376_mult__strict__mono,axiom,
    ! [C_57: quickcheck_code_int,D_10: quickcheck_code_int,A_135: quickcheck_code_int,B_104: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_135),B_104))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_57),D_10))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_104))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_57))
           => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_135),C_57)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_104),D_10))) ) ) ) ) ).

tff(fact_1377_mult__strict__mono,axiom,
    ! [C_57: rat,D_10: rat,A_135: rat,B_104: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_135),B_104))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_57),D_10))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_104))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_57))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_135),C_57)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_104),D_10))) ) ) ) ) ).

tff(fact_1378_mult__le__cancel__left__neg,axiom,
    ! [A: int,B: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ).

tff(fact_1379_mult__le__cancel__left__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A)) ) ) ).

tff(fact_1380_mult__le__cancel__left__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),A)) ) ) ).

tff(fact_1381_mult__le__cancel__left__pos,axiom,
    ! [A: int,B: int,C: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ) ).

tff(fact_1382_mult__le__cancel__left__pos,axiom,
    ! [A: real,B: real,C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ) ).

tff(fact_1383_mult__le__cancel__left__pos,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) ) ) ).

tff(fact_1384_mult__left__le__one__le,axiom,
    ! [Y_21: int,X_27: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_27))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_21))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_21),one_one_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_21),X_27)),X_27)) ) ) ) ).

tff(fact_1385_mult__left__le__one__le,axiom,
    ! [Y_21: real,X_27: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_27))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_21))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),one_one_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_21),X_27)),X_27)) ) ) ) ).

tff(fact_1386_mult__left__le__one__le,axiom,
    ! [Y_21: rat,X_27: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_27))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_21))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),one_one_rat))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_21),X_27)),X_27)) ) ) ) ).

tff(fact_1387_mult__right__le__one__le,axiom,
    ! [Y_20: int,X_26: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_26))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_20))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_20),one_one_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_26),Y_20)),X_26)) ) ) ) ).

tff(fact_1388_mult__right__le__one__le,axiom,
    ! [Y_20: real,X_26: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_26))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_20))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_20),one_one_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_26),Y_20)),X_26)) ) ) ) ).

tff(fact_1389_mult__right__le__one__le,axiom,
    ! [Y_20: rat,X_26: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_26))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_20))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_20),one_one_rat))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_26),Y_20)),X_26)) ) ) ) ).

tff(fact_1390_zero__less__two,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

tff(fact_1391_zero__less__two,axiom,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

tff(fact_1392_zero__less__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real))) ).

tff(fact_1393_zero__less__two,axiom,
    hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral))) ).

tff(fact_1394_zero__less__two,axiom,
    hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int))) ).

tff(fact_1395_zero__less__two,axiom,
    hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ).

tff(fact_1396_power__less__imp__less__base,axiom,
    ! [A_134: int,N_47: nat,B_103: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_134),N_47)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_103),N_47)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_103))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_134),B_103)) ) ) ).

tff(fact_1397_power__less__imp__less__base,axiom,
    ! [A_134: nat,N_47: nat,B_103: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_134),N_47)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_103),N_47)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_103))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_134),B_103)) ) ) ).

tff(fact_1398_power__less__imp__less__base,axiom,
    ! [A_134: real,N_47: nat,B_103: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_134),N_47)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_103),N_47)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_103))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_134),B_103)) ) ) ).

tff(fact_1399_power__less__imp__less__base,axiom,
    ! [A_134: code_code_numeral,N_47: nat,B_103: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_134),N_47)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_103),N_47)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_103))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_134),B_103)) ) ) ).

tff(fact_1400_power__less__imp__less__base,axiom,
    ! [A_134: quickcheck_code_int,N_47: nat,B_103: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_134),N_47)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_103),N_47)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_103))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_134),B_103)) ) ) ).

tff(fact_1401_power__less__imp__less__base,axiom,
    ! [A_134: rat,N_47: nat,B_103: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_134),N_47)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_103),N_47)))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_103))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_134),B_103)) ) ) ).

tff(fact_1402_power__decreasing,axiom,
    ! [A_133: int,N_46: nat,N_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_133))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_133),one_one_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_133),N_45)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_133),N_46))) ) ) ) ).

tff(fact_1403_power__decreasing,axiom,
    ! [A_133: nat,N_46: nat,N_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_133))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_133),one_one_nat))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_133),N_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_133),N_46))) ) ) ) ).

tff(fact_1404_power__decreasing,axiom,
    ! [A_133: real,N_46: nat,N_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_133))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_133),one_one_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_133),N_45)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_133),N_46))) ) ) ) ).

tff(fact_1405_power__decreasing,axiom,
    ! [A_133: code_code_numeral,N_46: nat,N_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_133))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_133),one_on1645066479umeral))
         => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_133),N_45)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_133),N_46))) ) ) ) ).

tff(fact_1406_power__decreasing,axiom,
    ! [A_133: quickcheck_code_int,N_46: nat,N_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_133))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_133),one_on1684967323de_int))
         => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_133),N_45)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_133),N_46))) ) ) ) ).

tff(fact_1407_power__decreasing,axiom,
    ! [A_133: rat,N_46: nat,N_45: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_133))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_133),one_one_rat))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_133),N_45)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_133),N_46))) ) ) ) ).

tff(fact_1408_power__le__imp__le__exp,axiom,
    ! [M_19: nat,N_44: nat,A_132: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_132))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_132),M_19)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_132),N_44)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44)) ) ) ).

tff(fact_1409_power__le__imp__le__exp,axiom,
    ! [M_19: nat,N_44: nat,A_132: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_132))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_132),M_19)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_132),N_44)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44)) ) ) ).

tff(fact_1410_power__le__imp__le__exp,axiom,
    ! [M_19: nat,N_44: nat,A_132: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_132))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_132),M_19)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_132),N_44)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44)) ) ) ).

tff(fact_1411_power__le__imp__le__exp,axiom,
    ! [M_19: nat,N_44: nat,A_132: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_132))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_132),M_19)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_132),N_44)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44)) ) ) ).

tff(fact_1412_power__le__imp__le__exp,axiom,
    ! [M_19: nat,N_44: nat,A_132: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_132))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_132),M_19)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_132),N_44)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44)) ) ) ).

tff(fact_1413_power__le__imp__le__exp,axiom,
    ! [M_19: nat,N_44: nat,A_132: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_132))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_132),M_19)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_132),N_44)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44)) ) ) ).

tff(fact_1414_power__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ) ).

tff(fact_1415_power__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ) ).

tff(fact_1416_power__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ) ).

tff(fact_1417_power__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_1)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ) ).

tff(fact_1418_power__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_1)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ) ).

tff(fact_1419_power__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_1)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ) ).

tff(fact_1420_power__less__power__Suc,axiom,
    ! [N_43: nat,A_131: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_131))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_131),N_43)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_131),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_131),N_43)))) ) ).

tff(fact_1421_power__less__power__Suc,axiom,
    ! [N_43: nat,A_131: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_131))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_131),N_43)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_131),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_131),N_43)))) ) ).

tff(fact_1422_power__less__power__Suc,axiom,
    ! [N_43: nat,A_131: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_131))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_131),N_43)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_131),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_131),N_43)))) ) ).

tff(fact_1423_power__less__power__Suc,axiom,
    ! [N_43: nat,A_131: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_131))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_131),N_43)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_131),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_131),N_43)))) ) ).

tff(fact_1424_power__less__power__Suc,axiom,
    ! [N_43: nat,A_131: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_131))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_131),N_43)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_131),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_131),N_43)))) ) ).

tff(fact_1425_power__less__power__Suc,axiom,
    ! [N_43: nat,A_131: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_131))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_131),N_43)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_131),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_131),N_43)))) ) ).

tff(fact_1426_power__gt1__lemma,axiom,
    ! [N_42: nat,A_130: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_130))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_130),N_42)))) ) ).

tff(fact_1427_power__gt1__lemma,axiom,
    ! [N_42: nat,A_130: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_130))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_130),N_42)))) ) ).

tff(fact_1428_power__gt1__lemma,axiom,
    ! [N_42: nat,A_130: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_130))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_130),N_42)))) ) ).

tff(fact_1429_power__gt1__lemma,axiom,
    ! [N_42: nat,A_130: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_130))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_130),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_130),N_42)))) ) ).

tff(fact_1430_power__gt1__lemma,axiom,
    ! [N_42: nat,A_130: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_130))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_130),N_42)))) ) ).

tff(fact_1431_power__gt1__lemma,axiom,
    ! [N_42: nat,A_130: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_130))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_130),N_42)))) ) ).

tff(fact_1432_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_41) = one_one_int ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_41) = zero_zero_int ) ) ) ).

tff(fact_1433_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_41) = one_one_nat ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_41) = zero_zero_nat ) ) ) ).

tff(fact_1434_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_41) = one_one_real ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_41) = zero_zero_real ) ) ) ).

tff(fact_1435_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_41) = one_on1645066479umeral ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_41) = zero_z126310315umeral ) ) ) ).

tff(fact_1436_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_41) = one_one_complex ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_41) = zero_zero_complex ) ) ) ).

tff(fact_1437_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_41) = one_on1684967323de_int ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_41) = zero_z891286103de_int ) ) ) ).

tff(fact_1438_power__0__left,axiom,
    ! [N_41: nat] :
      ( ( ( N_41 = zero_zero_nat )
       => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_41) = one_one_rat ) )
      & ( ( N_41 != zero_zero_nat )
       => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_41) = zero_zero_rat ) ) ) ).

tff(fact_1439_power__strict__increasing,axiom,
    ! [A_129: int,N_40: nat,N_39: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_129))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_129),N_40)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_129),N_39))) ) ) ).

tff(fact_1440_power__strict__increasing,axiom,
    ! [A_129: nat,N_40: nat,N_39: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_129))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_129),N_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_129),N_39))) ) ) ).

tff(fact_1441_power__strict__increasing,axiom,
    ! [A_129: real,N_40: nat,N_39: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_129))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_129),N_40)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_129),N_39))) ) ) ).

tff(fact_1442_power__strict__increasing,axiom,
    ! [A_129: code_code_numeral,N_40: nat,N_39: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_129))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_129),N_40)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_129),N_39))) ) ) ).

tff(fact_1443_power__strict__increasing,axiom,
    ! [A_129: quickcheck_code_int,N_40: nat,N_39: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_129))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_129),N_40)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_129),N_39))) ) ) ).

tff(fact_1444_power__strict__increasing,axiom,
    ! [A_129: rat,N_40: nat,N_39: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_129))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_129),N_40)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_129),N_39))) ) ) ).

tff(fact_1445_power__less__imp__less__exp,axiom,
    ! [M_18: nat,N_38: nat,A_128: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_128))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_128),M_18)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_128),N_38)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38)) ) ) ).

tff(fact_1446_power__less__imp__less__exp,axiom,
    ! [M_18: nat,N_38: nat,A_128: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_128))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_128),M_18)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_128),N_38)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38)) ) ) ).

tff(fact_1447_power__less__imp__less__exp,axiom,
    ! [M_18: nat,N_38: nat,A_128: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_128))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_128),M_18)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_128),N_38)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38)) ) ) ).

tff(fact_1448_power__less__imp__less__exp,axiom,
    ! [M_18: nat,N_38: nat,A_128: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_128))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_128),M_18)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_128),N_38)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38)) ) ) ).

tff(fact_1449_power__less__imp__less__exp,axiom,
    ! [M_18: nat,N_38: nat,A_128: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_128))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_128),M_18)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_128),N_38)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38)) ) ) ).

tff(fact_1450_power__less__imp__less__exp,axiom,
    ! [M_18: nat,N_38: nat,A_128: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_128))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_128),M_18)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_128),N_38)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38)) ) ) ).

tff(fact_1451_power__strict__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ) ).

tff(fact_1452_power__strict__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ) ).

tff(fact_1453_power__strict__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ) ).

tff(fact_1454_power__strict__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_1)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ) ).

tff(fact_1455_power__strict__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_1)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ) ).

tff(fact_1456_power__strict__increasing__iff,axiom,
    ! [X_1: nat,Y_2: nat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_1)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ) ).

tff(fact_1457_four__x__squared,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_1458_Int2_Ozcong__zero,axiom,
    ! [M: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M))
         => ( X = zero_zero_int ) ) ) ) ).

tff(fact_1459_zcong__zless__0,axiom,
    ! [M: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),M))
         => ( A_3 = zero_zero_int ) ) ) ) ).

tff(fact_1460_zcong__zless__imp__eq,axiom,
    ! [B_2: int,M: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),M))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
             => ( A_3 = B_2 ) ) ) ) ) ) ).

tff(fact_1461_zpower__zdvd__prop1,axiom,
    ! [P_3: int,Y: int,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N))) ) ) ).

tff(fact_1462_zcong__zpower__zmult,axiom,
    ! [Z_1: nat,X: int,Y: nat,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),one_one_int),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1))),one_one_int),P_3)) ) ).

tff(fact_1463_zcong__zmult__prop3,axiom,
    ! [Y: int,X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3))
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),zero_zero_int),P_3)) ) ) ) ).

tff(fact_1464_convex__bound__le,axiom,
    ! [V_9: int,U_2: int,Y_19: int,X_25: int,A_127: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_25),A_127))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_19),A_127))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_9))
           => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_9) = one_one_int )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_2),X_25)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_19))),A_127)) ) ) ) ) ) ).

tff(fact_1465_convex__bound__le,axiom,
    ! [V_9: real,U_2: real,Y_19: real,X_25: real,A_127: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_25),A_127))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_19),A_127))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_9))
           => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_9) = one_one_real )
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_2),X_25)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_19))),A_127)) ) ) ) ) ) ).

tff(fact_1466_convex__bound__le,axiom,
    ! [V_9: rat,U_2: rat,Y_19: rat,X_25: rat,A_127: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_25),A_127))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_19),A_127))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_2))
         => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_9))
           => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_9) = one_one_rat )
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_2),X_25)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_19))),A_127)) ) ) ) ) ) ).

tff(fact_1467_power__Suc__less,axiom,
    ! [N_37: nat,A_126: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_126))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_126),one_one_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_126),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_126),N_37))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_126),N_37))) ) ) ).

tff(fact_1468_power__Suc__less,axiom,
    ! [N_37: nat,A_126: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_126))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_126),one_one_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_126),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_126),N_37))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_126),N_37))) ) ) ).

tff(fact_1469_power__Suc__less,axiom,
    ! [N_37: nat,A_126: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_126))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_126),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_126),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_126),N_37))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_126),N_37))) ) ) ).

tff(fact_1470_power__Suc__less,axiom,
    ! [N_37: nat,A_126: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_126))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_126),one_on1645066479umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_126),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_126),N_37))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_126),N_37))) ) ) ).

tff(fact_1471_power__Suc__less,axiom,
    ! [N_37: nat,A_126: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_126))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_126),one_on1684967323de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_126),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_126),N_37))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_126),N_37))) ) ) ).

tff(fact_1472_power__Suc__less,axiom,
    ! [N_37: nat,A_126: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_126))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_126),one_one_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_126),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_126),N_37))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_126),N_37))) ) ) ).

tff(fact_1473_power__strict__decreasing,axiom,
    ! [A_125: int,N_36: nat,N_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_125))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_125),one_one_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_125),N_35)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_125),N_36))) ) ) ) ).

tff(fact_1474_power__strict__decreasing,axiom,
    ! [A_125: nat,N_36: nat,N_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_125))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_125),one_one_nat))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_125),N_35)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_125),N_36))) ) ) ) ).

tff(fact_1475_power__strict__decreasing,axiom,
    ! [A_125: real,N_36: nat,N_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_125))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_125),one_one_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_125),N_35)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_125),N_36))) ) ) ) ).

tff(fact_1476_power__strict__decreasing,axiom,
    ! [A_125: code_code_numeral,N_36: nat,N_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_125))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_125),one_on1645066479umeral))
         => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_125),N_35)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_125),N_36))) ) ) ) ).

tff(fact_1477_power__strict__decreasing,axiom,
    ! [A_125: quickcheck_code_int,N_36: nat,N_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_125))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_125),one_on1684967323de_int))
         => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_125),N_35)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_125),N_36))) ) ) ) ).

tff(fact_1478_power__strict__decreasing,axiom,
    ! [A_125: rat,N_36: nat,N_35: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_125))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_125),one_one_rat))
         => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_125),N_35)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_125),N_36))) ) ) ) ).

tff(fact_1479_power__eq__imp__eq__base,axiom,
    ! [A_124: int,N_34: nat,B_102: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_124),N_34) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_102),N_34) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_124))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_102))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
           => ( A_124 = B_102 ) ) ) ) ) ).

tff(fact_1480_power__eq__imp__eq__base,axiom,
    ! [A_124: nat,N_34: nat,B_102: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_124),N_34) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_102),N_34) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_124))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_102))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
           => ( A_124 = B_102 ) ) ) ) ) ).

tff(fact_1481_power__eq__imp__eq__base,axiom,
    ! [A_124: real,N_34: nat,B_102: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_124),N_34) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_102),N_34) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_124))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_102))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
           => ( A_124 = B_102 ) ) ) ) ) ).

tff(fact_1482_power__eq__imp__eq__base,axiom,
    ! [A_124: code_code_numeral,N_34: nat,B_102: code_code_numeral] :
      ( ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_124),N_34) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_102),N_34) )
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_124))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_102))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
           => ( A_124 = B_102 ) ) ) ) ) ).

tff(fact_1483_power__eq__imp__eq__base,axiom,
    ! [A_124: quickcheck_code_int,N_34: nat,B_102: quickcheck_code_int] :
      ( ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_124),N_34) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_102),N_34) )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_124))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_102))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
           => ( A_124 = B_102 ) ) ) ) ) ).

tff(fact_1484_power__eq__imp__eq__base,axiom,
    ! [A_124: rat,N_34: nat,B_102: rat] :
      ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_124),N_34) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_102),N_34) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_124))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_102))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34))
           => ( A_124 = B_102 ) ) ) ) ) ).

tff(fact_1485_one__less__power,axiom,
    ! [N_33: nat,A_123: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_123),N_33))) ) ) ).

tff(fact_1486_one__less__power,axiom,
    ! [N_33: nat,A_123: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_123),N_33))) ) ) ).

tff(fact_1487_one__less__power,axiom,
    ! [N_33: nat,A_123: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_123),N_33))) ) ) ).

tff(fact_1488_one__less__power,axiom,
    ! [N_33: nat,A_123: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_123),N_33))) ) ) ).

tff(fact_1489_one__less__power,axiom,
    ! [N_33: nat,A_123: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_123),N_33))) ) ) ).

tff(fact_1490_one__less__power,axiom,
    ! [N_33: nat,A_123: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_123))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_123),N_33))) ) ) ).

tff(fact_1491_dvd__power,axiom,
    ! [X_24: real,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_one_real ) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_24),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_24),N_32))) ) ).

tff(fact_1492_dvd__power,axiom,
    ! [X_24: code_code_numeral,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_on1645066479umeral ) )
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_24),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_24),N_32))) ) ).

tff(fact_1493_dvd__power,axiom,
    ! [X_24: complex,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_one_complex ) )
     => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_24),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_24),N_32))) ) ).

tff(fact_1494_dvd__power,axiom,
    ! [X_24: quickcheck_code_int,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_on1684967323de_int ) )
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_24),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_24),N_32))) ) ).

tff(fact_1495_dvd__power,axiom,
    ! [X_24: rat,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_one_rat ) )
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_24),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_24),N_32))) ) ).

tff(fact_1496_dvd__power,axiom,
    ! [X_24: int,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_one_int ) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_24),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_24),N_32))) ) ).

tff(fact_1497_dvd__power,axiom,
    ! [X_24: nat,N_32: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
        | ( X_24 = one_one_nat ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_24),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_24),N_32))) ) ).

tff(fact_1498__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(number_number_of_int,min))) ).

tff(fact_1499__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)))) ).

tff(fact_1500_neg__one__power__eq__mod__m,axiom,
    ! [J_2: nat,K_2: nat,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)),M))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2) ) ) ) ).

tff(fact_1501_one__not__neg__one__mod__m,axiom,
    ! [M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(one_one_int),hAPP_int_int(number_number_of_int,min)),M)) ) ).

tff(fact_1502__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
    hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) ).

tff(fact_1503__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_,axiom,
    ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(number_number_of_int,min)))
   => ( hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)) != one_one_int ) ) ).

tff(fact_1504_Legendre__1mod4,axiom,
    ! [M: int] :
      ( hBOOL(hAPP_int_bool(zprime,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int)))
     => ( hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int)) = one_one_int ) ) ).

tff(fact_1505_le0,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

tff(fact_1506_number__of1,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,N))))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,N)))) ) ) ).

tff(fact_1507_less__zeroE,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_1508_zprime__factor__exists,axiom,
    ! [A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
     => ? [P_4: int] :
          ( hBOOL(hAPP_int_bool(zprime,P_4))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_3)) ) ) ).

tff(fact_1509_not__real__square__gt__zero,axiom,
    ! [X_1: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)))
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_1510_number__of__diff,axiom,
    ! [V_8: int,W_4: int] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_8)),hAPP_int_int(number_number_of_int,W_4)) ).

tff(fact_1511_number__of__diff,axiom,
    ! [V_8: int,W_4: int] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_8)),hAPP_int_real(number267125858f_real,W_4)) ).

tff(fact_1512_number__of__diff,axiom,
    ! [V_8: int,W_4: int] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_8)),hAPP_int_complex(number528085621omplex,W_4)) ).

tff(fact_1513_number__of__diff,axiom,
    ! [V_8: int,W_4: int] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_8)),hAPP_int_rat(number_number_of_rat,W_4)) ).

tff(fact_1514_dvd__reduce,axiom,
    ! [K_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) ) ).

tff(fact_1515_nat__dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat))
    <=> ( Ma = one_one_nat ) ) ).

tff(fact_1516_dvd__diff,axiom,
    ! [Z_8: real,X_23: real,Y_18: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_23),Y_18))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_23),Z_8))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_23),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_18),Z_8))) ) ) ).

tff(fact_1517_dvd__diff,axiom,
    ! [Z_8: complex,X_23: complex,Y_18: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_23),Y_18))
     => ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_23),Z_8))
       => hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_23),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_18),Z_8))) ) ) ).

tff(fact_1518_dvd__diff,axiom,
    ! [Z_8: rat,X_23: rat,Y_18: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_23),Y_18))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_23),Z_8))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_23),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_18),Z_8))) ) ) ).

tff(fact_1519_dvd__diff,axiom,
    ! [Z_8: int,X_23: int,Y_18: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_23),Y_18))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_23),Z_8))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_23),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_18),Z_8))) ) ) ).

tff(fact_1520_diff__bin__simps_I1_J,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls) = K_2 ).

tff(fact_1521_diff__bin__simps_I7_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).

tff(fact_1522_zdiff__zmult__distrib,axiom,
    ! [Z1: int,Z2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).

tff(fact_1523_zdiff__zmult__distrib2,axiom,
    ! [W: int,Z1: int,Z2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).

tff(fact_1524_Int2_Oaux1,axiom,
    ! [A_3: int,B_2: int,C_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = C_1 )
     => ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_2) ) ) ).

tff(fact_1525_zcong__zdiff,axiom,
    ! [C_1: int,D_3: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),D_3)),M)) ) ) ).

tff(fact_1526_zdvd__zdiffD,axiom,
    ! [K_2: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),N)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).

tff(fact_1527_nat__dvd__not__less,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) ) ) ).

tff(fact_1528_inf__period_I3_J,axiom,
    ! [Ta: real,D: real,D_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1))
     => ! [X_2: real,K: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Ta)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))) ) ) ).

tff(fact_1529_inf__period_I3_J,axiom,
    ! [Ta: complex,D: complex,D_1: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1))
     => ! [X_2: complex,K: complex] :
          ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),Ta)))
        <=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))) ) ) ).

tff(fact_1530_inf__period_I3_J,axiom,
    ! [Ta: rat,D: rat,D_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1))
     => ! [X_2: rat,K: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Ta)))
        <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))) ) ) ).

tff(fact_1531_inf__period_I3_J,axiom,
    ! [Ta: int,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int,K: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) ) ) ).

tff(fact_1532_inf__period_I4_J,axiom,
    ! [Ta: real,D: real,D_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1))
     => ! [X_2: real,K: real] :
          ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Ta)))
        <=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))) ) ) ).

tff(fact_1533_inf__period_I4_J,axiom,
    ! [Ta: complex,D: complex,D_1: complex] :
      ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1))
     => ! [X_2: complex,K: complex] :
          ( ~ hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),Ta)))
        <=> ~ hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))) ) ) ).

tff(fact_1534_inf__period_I4_J,axiom,
    ! [Ta: rat,D: rat,D_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1))
     => ! [X_2: rat,K: rat] :
          ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Ta)))
        <=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))) ) ) ).

tff(fact_1535_inf__period_I4_J,axiom,
    ! [Ta: int,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int,K: int] :
          ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
        <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) ) ) ).

tff(fact_1536_eq__add__iff2,axiom,
    ! [A: int,E_2: int,C: int,B: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D) )
    <=> ( C = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D) ) ) ).

tff(fact_1537_eq__add__iff2,axiom,
    ! [A: real,E_2: real,C: real,B: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D) )
    <=> ( C = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D) ) ) ).

tff(fact_1538_eq__add__iff2,axiom,
    ! [A: complex,E_2: complex,C: complex,B: complex,D: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D) )
    <=> ( C = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B),A)),E_2)),D) ) ) ).

tff(fact_1539_eq__add__iff2,axiom,
    ! [A: rat,E_2: rat,C: rat,B: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D) )
    <=> ( C = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D) ) ) ).

tff(fact_1540_eq__add__iff1,axiom,
    ! [A: int,E_2: int,C: int,B: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D) )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C) = D ) ) ).

tff(fact_1541_eq__add__iff1,axiom,
    ! [A: real,E_2: real,C: real,B: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C) = D ) ) ).

tff(fact_1542_eq__add__iff1,axiom,
    ! [A: complex,E_2: complex,C: complex,B: complex,D: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D) )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B)),E_2)),C) = D ) ) ).

tff(fact_1543_eq__add__iff1,axiom,
    ! [A: rat,E_2: rat,C: rat,B: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D) )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C) = D ) ) ).

tff(fact_1544_right__diff__distrib__number__of,axiom,
    ! [V_7: int,B_101: int,C_56: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_101),C_56)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),B_101)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),C_56)) ).

tff(fact_1545_right__diff__distrib__number__of,axiom,
    ! [V_7: int,B_101: real,C_56: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_101),C_56)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),B_101)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),C_56)) ).

tff(fact_1546_right__diff__distrib__number__of,axiom,
    ! [V_7: int,B_101: complex,C_56: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_101),C_56)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),B_101)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),C_56)) ).

tff(fact_1547_right__diff__distrib__number__of,axiom,
    ! [V_7: int,B_101: rat,C_56: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_101),C_56)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),B_101)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),C_56)) ).

tff(fact_1548_left__diff__distrib__number__of,axiom,
    ! [A_122: int,B_100: int,V_6: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_122),B_100)),hAPP_int_int(number_number_of_int,V_6)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_122),hAPP_int_int(number_number_of_int,V_6))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_100),hAPP_int_int(number_number_of_int,V_6))) ).

tff(fact_1549_left__diff__distrib__number__of,axiom,
    ! [A_122: real,B_100: real,V_6: int] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_122),B_100)),hAPP_int_real(number267125858f_real,V_6)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_122),hAPP_int_real(number267125858f_real,V_6))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_100),hAPP_int_real(number267125858f_real,V_6))) ).

tff(fact_1550_left__diff__distrib__number__of,axiom,
    ! [A_122: complex,B_100: complex,V_6: int] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_122),B_100)),hAPP_int_complex(number528085621omplex,V_6)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_122),hAPP_int_complex(number528085621omplex,V_6))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_100),hAPP_int_complex(number528085621omplex,V_6))) ).

tff(fact_1551_left__diff__distrib__number__of,axiom,
    ! [A_122: rat,B_100: rat,V_6: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_122),B_100)),hAPP_int_rat(number_number_of_rat,V_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_122),hAPP_int_rat(number_number_of_rat,V_6))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_100),hAPP_int_rat(number_number_of_rat,V_6))) ).

tff(fact_1552_diff__bin__simps_I9_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).

tff(fact_1553_diff__bin__simps_I10_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) ).

tff(fact_1554_diff__bin__simps_I3_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),L_2)) ).

tff(fact_1555_less__bin__lemma,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)) ) ).

tff(fact_1556_dvd__imp__le,axiom,
    ! [K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) ) ) ).

tff(fact_1557_dvd__mult__cancel,axiom,
    ! [K_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) ) ) ).

tff(fact_1558_xzgcda__linear__aux1,axiom,
    ! [A_3: int,R_3: int,B_2: int,M: int,C_1: int,D_3: int,N: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),B_2))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),D_3))),N)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),N))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_3),N)))) ).

tff(fact_1559_zcong__def,axiom,
    ! [A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B))) ) ).

tff(fact_1560_le__add__iff1,axiom,
    ! [A: int,E_2: int,C: int,B: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)) ) ).

tff(fact_1561_le__add__iff1,axiom,
    ! [A: real,E_2: real,C: real,B: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)) ) ).

tff(fact_1562_le__add__iff1,axiom,
    ! [A: rat,E_2: rat,C: rat,B: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)) ) ).

tff(fact_1563_le__add__iff2,axiom,
    ! [A: int,E_2: int,C: int,B: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))) ) ).

tff(fact_1564_le__add__iff2,axiom,
    ! [A: real,E_2: real,C: real,B: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))) ) ).

tff(fact_1565_le__add__iff2,axiom,
    ! [A: rat,E_2: rat,C: rat,B: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) ) ).

tff(fact_1566_less__add__iff2,axiom,
    ! [A: int,E_2: int,C: int,B: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))) ) ).

tff(fact_1567_less__add__iff2,axiom,
    ! [A: real,E_2: real,C: real,B: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))) ) ).

tff(fact_1568_less__add__iff2,axiom,
    ! [A: rat,E_2: rat,C: rat,B: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) ) ).

tff(fact_1569_less__add__iff1,axiom,
    ! [A: int,E_2: int,C: int,B: int,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)) ) ).

tff(fact_1570_less__add__iff1,axiom,
    ! [A: real,E_2: real,C: real,B: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)) ) ).

tff(fact_1571_less__add__iff1,axiom,
    ! [A: rat,E_2: rat,C: rat,B: rat,D: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)) ) ).

tff(fact_1572_dvd__mult__cancel1,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1)),Ma))
      <=> ( N_1 = one_one_nat ) ) ) ).

tff(fact_1573_dvd__mult__cancel2,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Ma)),Ma))
      <=> ( N_1 = one_one_nat ) ) ) ).

tff(fact_1574_add__number__of__diff1,axiom,
    ! [V_5: int,W_3: int,C_55: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_5)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_3)),C_55)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) ).

tff(fact_1575_add__number__of__diff1,axiom,
    ! [V_5: int,W_3: int,C_55: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_5)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_3)),C_55)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) ).

tff(fact_1576_add__number__of__diff1,axiom,
    ! [V_5: int,W_3: int,C_55: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_5)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_3)),C_55)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) ).

tff(fact_1577_add__number__of__diff1,axiom,
    ! [V_5: int,W_3: int,C_55: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_3)),C_55)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) ).

tff(fact_1578_Euler_Oaux1,axiom,
    ! [A_3: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A_3))
       => ( ( X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int))) ) ) ) ).

tff(fact_1579_zle__diff1__eq,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

tff(fact_1580_diff__bin__simps_I4_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).

tff(fact_1581_diff__bin__simps_I5_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).

tff(fact_1582_diff__bin__simps_I6_J,axiom,
    ! [L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) ).

tff(fact_1583_inv__not__p__minus__1__aux,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))),one_one_int),P_5))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),P_5)) ) ).

tff(fact_1584_less__not__refl,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

tff(fact_1585_nat__neq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( Ma != N_1 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) ) ) ).

tff(fact_1586_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).

tff(fact_1587_less__irrefl__nat,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

tff(fact_1588_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
     => ( M != N ) ) ).

tff(fact_1589_less__not__refl3,axiom,
    ! [S_1: nat,T_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1))
     => ( S_1 != T_1 ) ) ).

tff(fact_1590_nat__less__cases,axiom,
    ! [P_1: fun_nat_fun_nat_bool,Ma: nat,N_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma)) )
     => ( ( ( Ma = N_1 )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma)) )
       => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma)) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma)) ) ) ) ).

tff(fact_1591_nat__add__commute,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M) ).

tff(fact_1592_nat__add__left__commute,axiom,
    ! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z_1)) ).

tff(fact_1593_nat__add__assoc,axiom,
    ! [M: nat,N: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_2)) ).

tff(fact_1594_nat__add__left__cancel,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_1595_nat__add__right__cancel,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_1596_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
       => ( M = N ) ) ) ).

tff(fact_1597_le__trans,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2)) ) ) ).

tff(fact_1598_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1599_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).

tff(fact_1600_le__refl,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).

tff(fact_1601_nat__mult__commute,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) ).

tff(fact_1602_nat__mult__assoc,axiom,
    ! [M: nat,N: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2)) ).

tff(fact_1603_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ) ).

tff(fact_1604_zcong__square__zless,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3))
           => ( ( A_3 = one_one_int )
              | ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) ) ) ) ) ) ) ).

tff(fact_1605_zcong__square,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),one_one_int),P_3))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),P_3)) ) ) ) ) ).

tff(fact_1606_not__less0,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_1607_neq0__conv,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_1608_less__nat__zero__code,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

tff(fact_1609_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( N != zero_zero_nat ) ) ).

tff(fact_1610_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_1611_plus__nat_Oadd__0,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

tff(fact_1612_Nat_Oadd__0__right,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

tff(fact_1613_add__is__0,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        & ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_1614_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

tff(fact_1615_conj__le__cong,axiom,
    ! [P_2: bool,P_1: bool,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(P_1)
        <=> hBOOL(P_2) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(P_1) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(P_2) ) ) ) ).

tff(fact_1616_imp__le__cong,axiom,
    ! [P_2: bool,P_1: bool,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(P_1)
        <=> hBOOL(P_2) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(P_1) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(P_2) ) ) ) ).

tff(fact_1617_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

tff(fact_1618_le__0__eq,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),zero_zero_nat))
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_1619_mult__0,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

tff(fact_1620_mult__0__right,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

tff(fact_1621_mult__is__0,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_1622_mult__cancel1,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) )
    <=> ( ( Ma = N_1 )
        | ( K_1 = zero_zero_nat ) ) ) ).

tff(fact_1623_mult__cancel2,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1) )
    <=> ( ( Ma = N_1 )
        | ( K_1 = zero_zero_nat ) ) ) ).

tff(fact_1624_not__add__less1,axiom,
    ! [I: nat,J_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),I)) ).

tff(fact_1625_not__add__less2,axiom,
    ! [J_2: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),I)) ).

tff(fact_1626_nat__add__left__cancel__less,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_1627_trans__less__add1,axiom,
    ! [M: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))) ) ).

tff(fact_1628_trans__less__add2,axiom,
    ! [M: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))) ) ).

tff(fact_1629_add__less__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))) ) ).

tff(fact_1630_add__less__mono,axiom,
    ! [K_2: nat,L_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2))) ) ) ).

tff(fact_1631_less__add__eq__less,axiom,
    ! [M: nat,N: nat,K_2: nat,L_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_1632_add__lessD1,axiom,
    ! [I: nat,J_2: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) ) ).

tff(fact_1633_nat__less__le,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))
        & ( Ma != N_1 ) ) ) ).

tff(fact_1634_le__eq__less__or__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
        | ( Ma = N_1 ) ) ) ).

tff(fact_1635_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1636_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( ( M != N )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).

tff(fact_1637_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
        | ( M = N ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1638_zspecial__product,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_1639_le__add2,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

tff(fact_1640_le__add1,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) ).

tff(fact_1641_le__iff__add,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))
    <=> ? [K: nat] : N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K) ) ).

tff(fact_1642_nat__add__left__cancel__le,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_1643_trans__le__add1,axiom,
    ! [M: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))) ) ).

tff(fact_1644_trans__le__add2,axiom,
    ! [M: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))) ) ).

tff(fact_1645_add__le__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))) ) ).

tff(fact_1646_add__le__mono,axiom,
    ! [K_2: nat,L_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2))) ) ) ).

tff(fact_1647_add__leD2,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) ) ).

tff(fact_1648_add__leD1,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_1649_add__leE,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) ) ) ).

tff(fact_1650_add__mult__distrib2,axiom,
    ! [K_2: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) ).

tff(fact_1651_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2)) ).

tff(fact_1652_le__square,axiom,
    ! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M))) ).

tff(fact_1653_le__cube,axiom,
    ! [M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) ).

tff(fact_1654_mult__le__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))) ) ).

tff(fact_1655_mult__le__mono2,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))) ) ).

tff(fact_1656_mult__le__mono,axiom,
    ! [K_2: nat,L_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),L_2))) ) ) ).

tff(fact_1657_nat__mult__1,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

tff(fact_1658_nat__1__eq__mult__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1) )
    <=> ( ( Ma = one_one_nat )
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_1659_nat__mult__1__right,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

tff(fact_1660_nat__mult__eq__1__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1) = one_one_nat )
    <=> ( ( Ma = one_one_nat )
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_1661_power2__diff,axiom,
    ! [X_22: int,Y_17: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).

tff(fact_1662_power2__diff,axiom,
    ! [X_22: real,Y_17: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).

tff(fact_1663_power2__diff,axiom,
    ! [X_22: complex,Y_17: complex] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).

tff(fact_1664_power2__diff,axiom,
    ! [X_22: rat,Y_17: rat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) ).

tff(fact_1665_zdiff__power2,axiom,
    ! [A_3: int,B_2: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_1666_zdiff__power3,axiom,
    ! [A_3: int,B_2: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_1667_add__gr__0,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ) ).

tff(fact_1668_nat__0__less__mult__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ) ).

tff(fact_1669_mult__less__cancel1,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ) ).

tff(fact_1670_mult__less__cancel2,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ) ).

tff(fact_1671_mult__less__mono1,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))) ) ) ).

tff(fact_1672_mult__less__mono2,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))) ) ) ).

tff(fact_1673_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

tff(fact_1674_zdvd__mono,axiom,
    ! [Ma: int,Ta: int,K_1: int] :
      ( ( K_1 != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ta))) ) ) ).

tff(fact_1675_unity__coeff__ex,axiom,
    ! [P_1: fun_Co1584449344l_bool,L: code_code_numeral] :
      ( ? [X_2: code_code_numeral] : hBOOL(hAPP_C24501650l_bool(P_1,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,L),X_2)))
    <=> ? [X_2: code_code_numeral] :
          ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,L),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),zero_z126310315umeral)))
          & hBOOL(hAPP_C24501650l_bool(P_1,X_2)) ) ) ).

tff(fact_1676_unity__coeff__ex,axiom,
    ! [P_1: fun_real_bool,L: real] :
      ( ? [X_2: real] : hBOOL(hAPP_real_bool(P_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,L),X_2)))
    <=> ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),zero_zero_real)))
          & hBOOL(hAPP_real_bool(P_1,X_2)) ) ) ).

tff(fact_1677_unity__coeff__ex,axiom,
    ! [P_1: fun_complex_bool,L: complex] :
      ( ? [X_2: complex] : hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X_2)))
    <=> ? [X_2: complex] :
          ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,L),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),zero_zero_complex)))
          & hBOOL(hAPP_complex_bool(P_1,X_2)) ) ) ).

tff(fact_1678_unity__coeff__ex,axiom,
    ! [P_1: fun_Qu817627884t_bool,L: quickcheck_code_int] :
      ( ? [X_2: quickcheck_code_int] : hBOOL(hAPP_Q2096512830t_bool(P_1,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,L),X_2)))
    <=> ? [X_2: quickcheck_code_int] :
          ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,L),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),zero_z891286103de_int)))
          & hBOOL(hAPP_Q2096512830t_bool(P_1,X_2)) ) ) ).

tff(fact_1679_unity__coeff__ex,axiom,
    ! [P_1: fun_rat_bool,L: rat] :
      ( ? [X_2: rat] : hBOOL(hAPP_rat_bool(P_1,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,L),X_2)))
    <=> ? [X_2: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),zero_zero_rat)))
          & hBOOL(hAPP_rat_bool(P_1,X_2)) ) ) ).

tff(fact_1680_unity__coeff__ex,axiom,
    ! [P_1: fun_int_bool,L: int] :
      ( ? [X_2: int] : hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,L),X_2)))
    <=> ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),zero_zero_int)))
          & hBOOL(hAPP_int_bool(P_1,X_2)) ) ) ).

tff(fact_1681_unity__coeff__ex,axiom,
    ! [P_1: fun_nat_bool,L: nat] :
      ( ? [X_2: nat] : hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,L),X_2)))
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,L),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),zero_zero_nat)))
          & hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_1682_number__of2,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,pls))) ).

tff(fact_1683_mult__le__cancel1,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ) ).

tff(fact_1684_mult__le__cancel2,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ) ).

tff(fact_1685_neg__one__power,axiom,
    ! [N: nat] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N) = one_one_int )
      | ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N) = hAPP_int_int(number_number_of_int,min) ) ) ).

tff(fact_1686_Legendre__def,axiom,
    ! [A_3: int,P_3: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
       => ( hAPP_int_int(legendre(A_3),P_3) = zero_zero_int ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
           => ( hAPP_int_int(legendre(A_3),P_3) = one_one_int ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
           => ( hAPP_int_int(legendre(A_3),P_3) = hAPP_int_int(number_number_of_int,min) ) ) ) ) ) ).

tff(fact_1687_QuadRes__def,axiom,
    ! [Ma: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Ma),X_1))
    <=> ? [Y_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),Ma)) ) ).

tff(fact_1688_real__sum__squared__expand,axiom,
    ! [X: real,Y: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),Y)) ).

tff(fact_1689_divides__cases,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
     => ( ( M = zero_zero_nat )
        | ( M = N )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),M)) ) ) ).

tff(fact_1690_decr__mult__lemma,axiom,
    ! [K_1: int,P_1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
         => ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(P_1,X_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) ) ) ) ).

tff(fact_1691_ex__least__nat__less,axiom,
    ! [N_1: nat,P_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(P_1,N_1))
       => ? [K: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N_1))
            & ! [I_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K))
               => ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
            & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))) ) ) ) ).

tff(fact_1692_incr__mult__lemma,axiom,
    ! [K_1: int,P_1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
         => ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(P_1,X_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) ) ) ) ).

tff(fact_1693_two__realpow__ge__one,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) ).

tff(fact_1694_realpow__two__sum__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_1695_zcong__zless__unique,axiom,
    ! [A_3: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),M))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),X_2),M))
          & ! [Y_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),Y_1),M)) )
             => ( Y_1 = X_2 ) ) ) ) ).

tff(fact_1696_dvd_Oorder__refl,axiom,
    ! [X: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ).

tff(fact_1697_real__le__eq__diff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_2)),zero_zero_real)) ) ).

tff(fact_1698_dvd_Oless__irrefl,axiom,
    ! [X: nat] :
      ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ) ).

tff(fact_1699_dvd_Oeq__iff,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( X_1 = Y_2 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) ) ) ).

tff(fact_1700_dvd_Ole__less,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_1701_dvd_Oless__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_1702_dvd_Oless__le__not__le,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) ) ) ).

tff(fact_1703_divides__antisym,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_1704_dvd_Oneq__le__trans,axiom,
    ! [A_3: nat,B_2: nat] :
      ( ( A_3 != B_2 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) ) ) ) ).

tff(fact_1705_dvd_Oeq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).

tff(fact_1706_dvd_Oantisym__conv,axiom,
    ! [Y_2: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_1707_dvd_Ole__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
        | ( X = Y ) ) ) ).

tff(fact_1708_dvd_Ole__neq__trans,axiom,
    ! [A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
     => ( ( A_3 != B_2 )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) ) ) ) ).

tff(fact_1709_dvd_Oord__eq__le__trans,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( ( A_3 = B_2 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) ) ) ).

tff(fact_1710_dvd_Oord__le__eq__trans,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
     => ( ( B_2 = C_1 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) ) ) ).

tff(fact_1711_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
       => ( M = N ) ) ) ).

tff(fact_1712_dvd_Oantisym,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
       => ( X = Y ) ) ) ).

tff(fact_1713_dvd_Oorder__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) ) ) ).

tff(fact_1714_dvd__diff__nat,axiom,
    ! [N: nat,K_2: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

tff(fact_1715_dvd_Oord__eq__less__trans,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( ( A_3 = B_2 )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),B_2)) )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) ) ) ) ).

tff(fact_1716_dvd_Ole__less__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).

tff(fact_1717_dvd_Oless__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( X != Y ) ) ).

tff(fact_1718_dvd_Oless__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_1719_dvd_Oless__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).

tff(fact_1720_dvd_Oless__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_1721_dvd_Oless__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( X != Y ) ) ).

tff(fact_1722_dvd_Oless__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( Y != X ) ) ).

tff(fact_1723_dvd_Oless__imp__triv,axiom,
    ! [P_1: bool,X_1: nat,Y_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) )
       => hBOOL(P_1) ) ) ).

tff(fact_1724_dvd_Oord__less__eq__trans,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) )
     => ( ( B_2 = C_1 )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) ) ) ) ).

tff(fact_1725_dvd_Oless__le__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).

tff(fact_1726_dvd_Oless__asym_H,axiom,
    ! [A_3: nat,B_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) ) ) ).

tff(fact_1727_dvd_Oless__trans,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).

tff(fact_1728_dvd_Oless__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_1729_real__zero__not__eq__one,axiom,
    zero_zero_real != one_one_real ).

tff(fact_1730_real__le__antisym,axiom,
    ! [Z_1: real,W: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1))
       => ( Z_1 = W ) ) ) ).

tff(fact_1731_real__le__trans,axiom,
    ! [K_2: real,I: real,J_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K_2)) ) ) ).

tff(fact_1732_real__add__left__mono,axiom,
    ! [Z_1: real,X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),Y))) ) ).

tff(fact_1733_less__eq__real__def,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
        | ( X_1 = Y_2 ) ) ) ).

tff(fact_1734_real__less__def,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))
        & ( X_1 != Y_2 ) ) ) ).

tff(fact_1735_real__le__linear,axiom,
    ! [Z_1: real,W: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W))
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) ) ).

tff(fact_1736_real__le__refl,axiom,
    ! [W: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).

tff(fact_1737_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

tff(fact_1738_diff__self__eq__0,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

tff(fact_1739_minus__nat_Odiff__0,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

tff(fact_1740_diff__0__eq__0,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

tff(fact_1741_diff__less__mono2,axiom,
    ! [L_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))) ) ) ).

tff(fact_1742_less__imp__diff__less,axiom,
    ! [N: nat,J_2: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),N)),K_2)) ) ).

tff(fact_1743_diff__cancel2,axiom,
    ! [M: nat,K_2: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).

tff(fact_1744_Nat_Odiff__cancel,axiom,
    ! [K_2: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).

tff(fact_1745_diff__diff__left,axiom,
    ! [I: nat,J_2: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2)) ).

tff(fact_1746_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

tff(fact_1747_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

tff(fact_1748_Nat_Odiff__le__self,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).

tff(fact_1749_diff__le__mono2,axiom,
    ! [L_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))) ) ).

tff(fact_1750_diff__le__mono,axiom,
    ! [L_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L_2))) ) ).

tff(fact_1751_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

tff(fact_1752_eq__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1) )
        <=> ( Ma = N_1 ) ) ) ) ).

tff(fact_1753_Nat_Odiff__diff__eq,axiom,
    ! [N: nat,K_2: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

tff(fact_1754_le__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ) ) ).

tff(fact_1755_dvd__diffD,axiom,
    ! [K_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ) ) ).

tff(fact_1756_dvd__diffD1,axiom,
    ! [K_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) ) ) ) ).

tff(fact_1757_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2)) ).

tff(fact_1758_diff__mult__distrib2,axiom,
    ! [K_2: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) ).

tff(fact_1759_diff__less,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).

tff(fact_1760_zero__less__diff,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),Ma)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_1761_diff__add__0,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

tff(fact_1762_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

tff(fact_1763_diff__is__0__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N_1) = zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_1764_add__diff__inverse,axiom,
    ! [M: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

tff(fact_1765_less__diff__conv,axiom,
    ! [I_1: nat,J: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) ) ).

tff(fact_1766_less__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ) ) ).

tff(fact_1767_diff__less__mono,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_3),B_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C_1))) ) ) ).

tff(fact_1768_diff__add__assoc2,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) ) ) ).

tff(fact_1769_add__diff__assoc2,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2) ) ) ).

tff(fact_1770_diff__add__assoc,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) ) ) ).

tff(fact_1771_le__imp__diff__is__add,axiom,
    ! [K_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1) = K_1 )
      <=> ( J = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) ) ) ) ).

tff(fact_1772_le__add__diff__inverse2,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = M ) ) ).

tff(fact_1773_le__diff__conv2,axiom,
    ! [I_1: nat,K_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) ) ) ).

tff(fact_1774_add__diff__assoc,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) ) ) ).

tff(fact_1775_le__add__diff__inverse,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

tff(fact_1776_le__add__diff,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),K_2))) ) ).

tff(fact_1777_le__diff__conv,axiom,
    ! [J: nat,K_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))) ) ).

tff(fact_1778_diff__diff__right,axiom,
    ! [I: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),J_2) ) ) ).

tff(fact_1779_nat__diff__split,axiom,
    ! [P_1: fun_nat_bool,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
        & ! [D_2: nat] :
            ( ( A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2) )
           => hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).

tff(fact_1780_nat__diff__split__asm,axiom,
    ! [P_1: fun_nat_bool,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
            & ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
          | ? [D_2: nat] :
              ( ( A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2) )
              & ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).

tff(fact_1781_divides__add__revr,axiom,
    ! [B_2: nat,D_3: nat,A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2)) ) ) ).

tff(fact_1782_real__mult__1,axiom,
    ! [Z_1: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1) = Z_1 ).

tff(fact_1783_real__mult__commute,axiom,
    ! [Z_1: real,W: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1) ).

tff(fact_1784_real__mult__assoc,axiom,
    ! [Z1: real,Z2: real,Z3: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),Z2)),Z3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),Z3)) ).

tff(fact_1785_real__add__mult__distrib,axiom,
    ! [Z1: real,Z2: real,W: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z1),Z2)),W) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),W)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),W)) ).

tff(fact_1786_real__two__squares__add__zero__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_1787_real__mult__right__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( C != zero_zero_real )
     => ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

tff(fact_1788_real__mult__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( C != zero_zero_real )
     => ( ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

tff(fact_1789_real__mult__less__iff1,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ).

tff(fact_1790_real__mult__order,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))) ) ) ).

tff(fact_1791_real__mult__less__mono2,axiom,
    ! [X: real,Y: real,Z_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),Y))) ) ) ).

tff(fact_1792_real__mult__le__cancel__iff2,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ).

tff(fact_1793_real__mult__le__cancel__iff1,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ).

tff(fact_1794_divides__mul__l,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),A_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),B_2))) ) ).

tff(fact_1795_divides__mul__r,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))) ) ).

tff(fact_1796_realpow__minus__mult,axiom,
    ! [X_21: code_code_numeral,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),N_31) ) ) ).

tff(fact_1797_realpow__minus__mult,axiom,
    ! [X_21: quickcheck_code_int,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),N_31) ) ) ).

tff(fact_1798_realpow__minus__mult,axiom,
    ! [X_21: rat,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),N_31) ) ) ).

tff(fact_1799_realpow__minus__mult,axiom,
    ! [X_21: int,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),N_31) ) ) ).

tff(fact_1800_realpow__minus__mult,axiom,
    ! [X_21: nat,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),N_31) ) ) ).

tff(fact_1801_realpow__minus__mult,axiom,
    ! [X_21: real,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),N_31) ) ) ).

tff(fact_1802_realpow__minus__mult,axiom,
    ! [X_21: complex,N_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
     => ( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),N_31) ) ) ).

tff(fact_1803_divides__exp,axiom,
    ! [N: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),N))) ) ).

tff(fact_1804_mult__eq__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

tff(fact_1805_power__eq__if,axiom,
    ! [P_3: nat,M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

tff(fact_1806_diff__square,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).

tff(fact_1807_add__diff__add,axiom,
    ! [A_121: int,C_54: int,B_99: int,D_9: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_121),C_54)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_99),D_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_121),B_99)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_54),D_9)) ).

tff(fact_1808_add__diff__add,axiom,
    ! [A_121: real,C_54: real,B_99: real,D_9: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_121),C_54)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_99),D_9)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_121),B_99)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_54),D_9)) ).

tff(fact_1809_add__diff__add,axiom,
    ! [A_121: complex,C_54: complex,B_99: complex,D_9: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_121),C_54)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_99),D_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_121),B_99)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_54),D_9)) ).

tff(fact_1810_add__diff__add,axiom,
    ! [A_121: rat,C_54: rat,B_99: rat,D_9: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_121),C_54)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_99),D_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_121),B_99)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_54),D_9)) ).

tff(fact_1811_divides__ge,axiom,
    ! [A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))
     => ( ( B_2 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),B_2)) ) ) ).

tff(fact_1812_nat__mult__dvd__cancel__disj_H,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)))
    <=> ( ( K_1 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N_1)) ) ) ).

tff(fact_1813_nat__mult__eq__one,axiom,
    ! [N_1: nat,Ma: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Ma) = one_one_nat )
    <=> ( ( N_1 = one_one_nat )
        & ( Ma = one_one_nat ) ) ) ).

tff(fact_1814_nat__power__eq__0__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ma),N_1) = zero_zero_nat )
    <=> ( ( N_1 != zero_zero_nat )
        & ( Ma = zero_zero_nat ) ) ) ).

tff(fact_1815_divides__rev,axiom,
    ! [A_3: nat,N: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N)))
     => ( ( N != zero_zero_nat )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) ) ) ).

tff(fact_1816_divides__exp2,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( N != zero_zero_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).

tff(fact_1817_mult__diff__mult,axiom,
    ! [X_20: int,Y_16: int,A_120: int,B_98: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_20),Y_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_120),B_98)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_20),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_16),B_98))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_20),A_120)),B_98)) ).

tff(fact_1818_mult__diff__mult,axiom,
    ! [X_20: real,Y_16: real,A_120: real,B_98: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_20),Y_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_120),B_98)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_20),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_16),B_98))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_20),A_120)),B_98)) ).

tff(fact_1819_mult__diff__mult,axiom,
    ! [X_20: complex,Y_16: complex,A_120: complex,B_98: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_20),Y_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_120),B_98)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_20),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_16),B_98))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_20),A_120)),B_98)) ).

tff(fact_1820_mult__diff__mult,axiom,
    ! [X_20: rat,Y_16: rat,A_120: rat,B_98: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_20),Y_16)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_120),B_98)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_20),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_16),B_98))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_20),A_120)),B_98)) ).

tff(fact_1821_exp__eq__1,axiom,
    ! [X_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1) = one_one_nat )
    <=> ( ( X_1 = one_one_nat )
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_1822_real__squared__diff__one__factored,axiom,
    ! [X_19: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_19),X_19)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_19),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_19),one_one_int)) ).

tff(fact_1823_real__squared__diff__one__factored,axiom,
    ! [X_19: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_19),X_19)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_19),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_19),one_one_real)) ).

tff(fact_1824_real__squared__diff__one__factored,axiom,
    ! [X_19: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_19),X_19)),one_one_complex) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_19),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_19),one_one_complex)) ).

tff(fact_1825_real__squared__diff__one__factored,axiom,
    ! [X_19: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_19),X_19)),one_one_rat) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_19),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_19),one_one_rat)) ).

tff(fact_1826_divides__div__not,axiom,
    ! [X: nat,Q_2: nat,N: nat,R_3: nat] :
      ( ( X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_2),N)),R_3) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_3))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),N))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)) ) ) ) ).

tff(fact_1827_realpow__pos__nth__unique,axiom,
    ! [A_3: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
       => ? [X_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
            & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N) = A_3 )
            & ! [Y_1: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
                  & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N) = A_3 ) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_1828_realpow__pos__nth,axiom,
    ! [A_3: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N) = A_3 ) ) ) ) ).

tff(fact_1829_realpow__num__eq__if,axiom,
    ! [M_17: int,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30) = one_one_int ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_17),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1830_realpow__num__eq__if,axiom,
    ! [M_17: nat,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = one_one_nat ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1831_realpow__num__eq__if,axiom,
    ! [M_17: real,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30) = one_one_real ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_17),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1832_realpow__num__eq__if,axiom,
    ! [M_17: code_code_numeral,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = one_on1645066479umeral ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_17),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1833_realpow__num__eq__if,axiom,
    ! [M_17: complex,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30) = one_one_complex ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,M_17),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1834_realpow__num__eq__if,axiom,
    ! [M_17: quickcheck_code_int,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = one_on1684967323de_int ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_17),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1835_realpow__num__eq__if,axiom,
    ! [M_17: rat,N_30: nat] :
      ( ( ( N_30 = zero_zero_nat )
       => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30) = one_one_rat ) )
      & ( ( N_30 != zero_zero_nat )
       => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_17),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) ) ) ) ).

tff(fact_1836__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
    hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,s),one_one_int)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),t) ).

tff(fact_1837_norR__mem__unique__aux,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) ) ).

tff(fact_1838_nat__less__add__iff1,axiom,
    ! [U: nat,Ma: nat,N_1: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N_1)) ) ) ).

tff(fact_1839_nat__less__add__iff2,axiom,
    ! [U: nat,Ma: nat,N_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N_1))) ) ) ).

tff(fact_1840_nat__mult__le__cancel1,axiom,
    ! [Ma: nat,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ) ).

tff(fact_1841_nat__mult__dvd__cancel1,axiom,
    ! [Ma: nat,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N_1)) ) ) ).

tff(fact_1842_diff__commute,axiom,
    ! [I: nat,J_2: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K_2)),J_2) ).

tff(fact_1843_mult__sum2sq,axiom,
    ! [A_3: int,B_2: int,P_3: int,Q_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2))),hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2))) = hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),P_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),P_3)))) ).

tff(fact_1844_nat__mult__eq__cancel__disj,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) )
    <=> ( ( K_1 = zero_zero_nat )
        | ( Ma = N_1 ) ) ) ).

tff(fact_1845_left__add__mult__distrib,axiom,
    ! [I: nat,U_1: nat,J_2: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),U_1)),K_2) ).

tff(fact_1846_nat__mult__eq__cancel1,axiom,
    ! [Ma: nat,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) )
      <=> ( Ma = N_1 ) ) ) ).

tff(fact_1847_nat__mult__less__cancel1,axiom,
    ! [Ma: nat,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ) ).

tff(fact_1848_nat__mult__dvd__cancel__disj,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
    <=> ( ( K_1 = zero_zero_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N_1)) ) ) ).

tff(fact_1849_nat__eq__add__iff2,axiom,
    ! [U: nat,Ma: nat,N_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1) )
      <=> ( Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N_1) ) ) ) ).

tff(fact_1850_nat__diff__add__eq2,axiom,
    ! [U_1: nat,M: nat,N: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I)),U_1)),N)) ) ) ).

tff(fact_1851_nat__le__add__iff2,axiom,
    ! [U: nat,Ma: nat,N_1: nat,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N_1))) ) ) ).

tff(fact_1852_nat__eq__add__iff1,axiom,
    ! [U: nat,Ma: nat,N_1: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1) )
      <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma) = N_1 ) ) ) ).

tff(fact_1853_nat__diff__add__eq1,axiom,
    ! [U_1: nat,M: nat,N: nat,J_2: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),U_1)),M)),N) ) ) ).

tff(fact_1854_nat__le__add__iff1,axiom,
    ! [U: nat,Ma: nat,N_1: nat,J: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N_1)) ) ) ).

tff(fact_1855_is__sum2sq__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(twoSqu1425020248sum2sq,X_1))
    <=> ? [A_2: int,B_4: int] : hAPP_P1175774780nt_int(twoSqu2000912462sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)) = X_1 ) ).

tff(fact_1856_Wilson__Russ,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_int(number_number_of_int,min)),P_3)) ) ).

tff(fact_1857_add__pos__nonneg,axiom,
    ! [B_97: code_code_numeral,A_119: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_119))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_97))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_119),B_97))) ) ) ).

tff(fact_1858_add__pos__nonneg,axiom,
    ! [B_97: int,A_119: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_119))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_97))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_119),B_97))) ) ) ).

tff(fact_1859_add__pos__nonneg,axiom,
    ! [B_97: nat,A_119: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_119))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_97))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_119),B_97))) ) ) ).

tff(fact_1860_add__pos__nonneg,axiom,
    ! [B_97: real,A_119: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_119))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_97))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_119),B_97))) ) ) ).

tff(fact_1861_add__pos__nonneg,axiom,
    ! [B_97: quickcheck_code_int,A_119: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_119))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_97))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_119),B_97))) ) ) ).

tff(fact_1862_add__pos__nonneg,axiom,
    ! [B_97: rat,A_119: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_119))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_97))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_119),B_97))) ) ) ).

tff(fact_1863_add__nonneg__pos,axiom,
    ! [B_96: code_code_numeral,A_118: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_118))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_96))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_118),B_96))) ) ) ).

tff(fact_1864_add__nonneg__pos,axiom,
    ! [B_96: int,A_118: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_118))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_96))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_118),B_96))) ) ) ).

tff(fact_1865_add__nonneg__pos,axiom,
    ! [B_96: nat,A_118: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_118))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_96))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_118),B_96))) ) ) ).

tff(fact_1866_add__nonneg__pos,axiom,
    ! [B_96: real,A_118: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_118))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_96))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_118),B_96))) ) ) ).

tff(fact_1867_add__nonneg__pos,axiom,
    ! [B_96: quickcheck_code_int,A_118: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_118))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_96))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_118),B_96))) ) ) ).

tff(fact_1868_add__nonneg__pos,axiom,
    ! [B_96: rat,A_118: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_118))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_96))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_118),B_96))) ) ) ).

tff(fact_1869_add__strict__increasing,axiom,
    ! [B_95: code_code_numeral,C_53: code_code_numeral,A_117: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_117))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_95),C_53))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_95),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_117),C_53))) ) ) ).

tff(fact_1870_add__strict__increasing,axiom,
    ! [B_95: int,C_53: int,A_117: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_117))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_95),C_53))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_95),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_117),C_53))) ) ) ).

tff(fact_1871_add__strict__increasing,axiom,
    ! [B_95: nat,C_53: nat,A_117: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_117))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_95),C_53))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_95),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_117),C_53))) ) ) ).

tff(fact_1872_add__strict__increasing,axiom,
    ! [B_95: real,C_53: real,A_117: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_117))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_95),C_53))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_95),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_117),C_53))) ) ) ).

tff(fact_1873_add__strict__increasing,axiom,
    ! [B_95: quickcheck_code_int,C_53: quickcheck_code_int,A_117: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_117))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_95),C_53))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_95),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_117),C_53))) ) ) ).

tff(fact_1874_add__strict__increasing,axiom,
    ! [B_95: rat,C_53: rat,A_117: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_117))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_95),C_53))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_95),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_117),C_53))) ) ) ).

tff(fact_1875_add__strict__increasing2,axiom,
    ! [B_94: code_code_numeral,C_52: code_code_numeral,A_116: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_116))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_94),C_52))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_94),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_116),C_52))) ) ) ).

tff(fact_1876_add__strict__increasing2,axiom,
    ! [B_94: int,C_52: int,A_116: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_116))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_94),C_52))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_94),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_116),C_52))) ) ) ).

tff(fact_1877_add__strict__increasing2,axiom,
    ! [B_94: nat,C_52: nat,A_116: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_116))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_94),C_52))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_94),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_116),C_52))) ) ) ).

tff(fact_1878_add__strict__increasing2,axiom,
    ! [B_94: real,C_52: real,A_116: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_116))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_94),C_52))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_94),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_116),C_52))) ) ) ).

tff(fact_1879_add__strict__increasing2,axiom,
    ! [B_94: quickcheck_code_int,C_52: quickcheck_code_int,A_116: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_116))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_94),C_52))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_94),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_116),C_52))) ) ) ).

tff(fact_1880_add__strict__increasing2,axiom,
    ! [B_94: rat,C_52: rat,A_116: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_116))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_94),C_52))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_94),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_116),C_52))) ) ) ).

tff(fact_1881_add__neg__nonpos,axiom,
    ! [B_93: code_code_numeral,A_115: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_115),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_93),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_115),B_93)),zero_z126310315umeral)) ) ) ).

tff(fact_1882_add__neg__nonpos,axiom,
    ! [B_93: int,A_115: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_115),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_93),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_115),B_93)),zero_zero_int)) ) ) ).

tff(fact_1883_add__neg__nonpos,axiom,
    ! [B_93: nat,A_115: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_115),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_93),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_115),B_93)),zero_zero_nat)) ) ) ).

tff(fact_1884_add__neg__nonpos,axiom,
    ! [B_93: real,A_115: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_115),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_93),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_115),B_93)),zero_zero_real)) ) ) ).

tff(fact_1885_add__neg__nonpos,axiom,
    ! [B_93: quickcheck_code_int,A_115: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_115),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_93),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_115),B_93)),zero_z891286103de_int)) ) ) ).

tff(fact_1886_add__neg__nonpos,axiom,
    ! [B_93: rat,A_115: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_115),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_93),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_115),B_93)),zero_zero_rat)) ) ) ).

tff(fact_1887_add__nonpos__neg,axiom,
    ! [B_92: code_code_numeral,A_114: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_114),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_92),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_114),B_92)),zero_z126310315umeral)) ) ) ).

tff(fact_1888_add__nonpos__neg,axiom,
    ! [B_92: int,A_114: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_114),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_92),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_114),B_92)),zero_zero_int)) ) ) ).

tff(fact_1889_add__nonpos__neg,axiom,
    ! [B_92: nat,A_114: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_114),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_92),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_114),B_92)),zero_zero_nat)) ) ) ).

tff(fact_1890_add__nonpos__neg,axiom,
    ! [B_92: real,A_114: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_114),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_92),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_114),B_92)),zero_zero_real)) ) ) ).

tff(fact_1891_add__nonpos__neg,axiom,
    ! [B_92: quickcheck_code_int,A_114: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_114),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_92),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_114),B_92)),zero_z891286103de_int)) ) ) ).

tff(fact_1892_add__nonpos__neg,axiom,
    ! [B_92: rat,A_114: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_114),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_92),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_114),B_92)),zero_zero_rat)) ) ) ).

tff(fact_1893_inv__inv,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3))
           => ( hAPP_int_int(inv(P_3),hAPP_int_int(inv(P_3),A_3)) = A_3 ) ) ) ) ) ).

tff(fact_1894_zero__reorient,axiom,
    ! [X_1: int] :
      ( ( zero_zero_int = X_1 )
    <=> ( X_1 = zero_zero_int ) ) ).

tff(fact_1895_zero__reorient,axiom,
    ! [X_1: nat] :
      ( ( zero_zero_nat = X_1 )
    <=> ( X_1 = zero_zero_nat ) ) ).

tff(fact_1896_zero__reorient,axiom,
    ! [X_1: real] :
      ( ( zero_zero_real = X_1 )
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_1897_zero__reorient,axiom,
    ! [X_1: code_code_numeral] :
      ( ( zero_z126310315umeral = X_1 )
    <=> ( X_1 = zero_z126310315umeral ) ) ).

tff(fact_1898_zero__reorient,axiom,
    ! [X_1: complex] :
      ( ( zero_zero_complex = X_1 )
    <=> ( X_1 = zero_zero_complex ) ) ).

tff(fact_1899_zero__reorient,axiom,
    ! [X_1: quickcheck_code_int] :
      ( ( zero_z891286103de_int = X_1 )
    <=> ( X_1 = zero_z891286103de_int ) ) ).

tff(fact_1900_zero__reorient,axiom,
    ! [X_1: rat] :
      ( ( zero_zero_rat = X_1 )
    <=> ( X_1 = zero_zero_rat ) ) ).

tff(fact_1901_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: code_code_numeral,B_91: code_code_numeral,C_51: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_113),B_91)),C_51) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_113),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_91),C_51)) ).

tff(fact_1902_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: int,B_91: int,C_51: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_113),B_91)),C_51) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_113),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_91),C_51)) ).

tff(fact_1903_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: nat,B_91: nat,C_51: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_113),B_91)),C_51) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_113),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_91),C_51)) ).

tff(fact_1904_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: real,B_91: real,C_51: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_113),B_91)),C_51) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_113),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_91),C_51)) ).

tff(fact_1905_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: complex,B_91: complex,C_51: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_113),B_91)),C_51) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_113),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_91),C_51)) ).

tff(fact_1906_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: quickcheck_code_int,B_91: quickcheck_code_int,C_51: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_113),B_91)),C_51) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_113),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_91),C_51)) ).

tff(fact_1907_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A_113: rat,B_91: rat,C_51: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_113),B_91)),C_51) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_113),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_91),C_51)) ).

tff(fact_1908_add__right__imp__eq,axiom,
    ! [B_90: code_code_numeral,A_112: code_code_numeral,C_50: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_90),A_112) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1909_add__right__imp__eq,axiom,
    ! [B_90: int,A_112: int,C_50: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_90),A_112) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1910_add__right__imp__eq,axiom,
    ! [B_90: nat,A_112: nat,C_50: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_90),A_112) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1911_add__right__imp__eq,axiom,
    ! [B_90: real,A_112: real,C_50: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_90),A_112) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1912_add__right__imp__eq,axiom,
    ! [B_90: complex,A_112: complex,C_50: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_90),A_112) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1913_add__right__imp__eq,axiom,
    ! [B_90: quickcheck_code_int,A_112: quickcheck_code_int,C_50: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_90),A_112) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1914_add__right__imp__eq,axiom,
    ! [B_90: rat,A_112: rat,C_50: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_90),A_112) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_50),A_112) )
     => ( B_90 = C_50 ) ) ).

tff(fact_1915_add__imp__eq,axiom,
    ! [A_111: code_code_numeral,B_89: code_code_numeral,C_49: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_111),B_89) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1916_add__imp__eq,axiom,
    ! [A_111: int,B_89: int,C_49: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_111),B_89) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1917_add__imp__eq,axiom,
    ! [A_111: nat,B_89: nat,C_49: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_111),B_89) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1918_add__imp__eq,axiom,
    ! [A_111: real,B_89: real,C_49: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_111),B_89) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1919_add__imp__eq,axiom,
    ! [A_111: complex,B_89: complex,C_49: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_111),B_89) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1920_add__imp__eq,axiom,
    ! [A_111: quickcheck_code_int,B_89: quickcheck_code_int,C_49: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_111),B_89) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1921_add__imp__eq,axiom,
    ! [A_111: rat,B_89: rat,C_49: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_111),B_89) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_111),C_49) )
     => ( B_89 = C_49 ) ) ).

tff(fact_1922_add__left__imp__eq,axiom,
    ! [A_110: code_code_numeral,B_88: code_code_numeral,C_48: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),B_88) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1923_add__left__imp__eq,axiom,
    ! [A_110: int,B_88: int,C_48: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),B_88) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1924_add__left__imp__eq,axiom,
    ! [A_110: nat,B_88: nat,C_48: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),B_88) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1925_add__left__imp__eq,axiom,
    ! [A_110: real,B_88: real,C_48: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),B_88) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1926_add__left__imp__eq,axiom,
    ! [A_110: complex,B_88: complex,C_48: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),B_88) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1927_add__left__imp__eq,axiom,
    ! [A_110: quickcheck_code_int,B_88: quickcheck_code_int,C_48: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),B_88) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1928_add__left__imp__eq,axiom,
    ! [A_110: rat,B_88: rat,C_48: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),B_88) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),C_48) )
     => ( B_88 = C_48 ) ) ).

tff(fact_1929_add__right__cancel,axiom,
    ! [B: code_code_numeral,A: code_code_numeral,C: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1930_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1931_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1932_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1933_add__right__cancel,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1934_add__right__cancel,axiom,
    ! [B: quickcheck_code_int,A: quickcheck_code_int,C: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),A) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1935_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

tff(fact_1936_add__left__cancel,axiom,
    ! [A: code_code_numeral,B: code_code_numeral,C: code_code_numeral] :
      ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),B) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1937_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1938_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),B) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1939_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1940_add__left__cancel,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),B) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1941_add__left__cancel,axiom,
    ! [A: quickcheck_code_int,B: quickcheck_code_int,C: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),B) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1942_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

tff(fact_1943_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: code_code_numeral,B_87: code_code_numeral,C_47: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),B_87)),C_47) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_87),C_47)) ).

tff(fact_1944_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: int,B_87: int,C_47: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),B_87)),C_47) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_87),C_47)) ).

tff(fact_1945_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: nat,B_87: nat,C_47: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),B_87)),C_47) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_87),C_47)) ).

tff(fact_1946_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: real,B_87: real,C_47: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),B_87)),C_47) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_87),C_47)) ).

tff(fact_1947_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: complex,B_87: complex,C_47: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),B_87)),C_47) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_87),C_47)) ).

tff(fact_1948_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: quickcheck_code_int,B_87: quickcheck_code_int,C_47: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),B_87)),C_47) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_87),C_47)) ).

tff(fact_1949_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_109: rat,B_87: rat,C_47: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),B_87)),C_47) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_87),C_47)) ).

tff(fact_1950_one__reorient,axiom,
    ! [X_1: int] :
      ( ( one_one_int = X_1 )
    <=> ( X_1 = one_one_int ) ) ).

tff(fact_1951_one__reorient,axiom,
    ! [X_1: nat] :
      ( ( one_one_nat = X_1 )
    <=> ( X_1 = one_one_nat ) ) ).

tff(fact_1952_one__reorient,axiom,
    ! [X_1: real] :
      ( ( one_one_real = X_1 )
    <=> ( X_1 = one_one_real ) ) ).

tff(fact_1953_one__reorient,axiom,
    ! [X_1: code_code_numeral] :
      ( ( one_on1645066479umeral = X_1 )
    <=> ( X_1 = one_on1645066479umeral ) ) ).

tff(fact_1954_one__reorient,axiom,
    ! [X_1: complex] :
      ( ( one_one_complex = X_1 )
    <=> ( X_1 = one_one_complex ) ) ).

tff(fact_1955_one__reorient,axiom,
    ! [X_1: quickcheck_code_int] :
      ( ( one_on1684967323de_int = X_1 )
    <=> ( X_1 = one_on1684967323de_int ) ) ).

tff(fact_1956_one__reorient,axiom,
    ! [X_1: rat] :
      ( ( one_one_rat = X_1 )
    <=> ( X_1 = one_one_rat ) ) ).

tff(fact_1957_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

tff(fact_1958_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

tff(fact_1959_diff__eq__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

tff(fact_1960_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

tff(fact_1961_add_Ocomm__neutral,axiom,
    ! [A_108: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_108),zero_z126310315umeral) = A_108 ).

tff(fact_1962_add_Ocomm__neutral,axiom,
    ! [A_108: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_108),zero_zero_int) = A_108 ).

tff(fact_1963_add_Ocomm__neutral,axiom,
    ! [A_108: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_108),zero_zero_nat) = A_108 ).

tff(fact_1964_add_Ocomm__neutral,axiom,
    ! [A_108: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_108),zero_zero_real) = A_108 ).

tff(fact_1965_add_Ocomm__neutral,axiom,
    ! [A_108: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_108),zero_zero_complex) = A_108 ).

tff(fact_1966_add_Ocomm__neutral,axiom,
    ! [A_108: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_108),zero_z891286103de_int) = A_108 ).

tff(fact_1967_add_Ocomm__neutral,axiom,
    ! [A_108: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_108),zero_zero_rat) = A_108 ).

tff(fact_1968_add__0__right,axiom,
    ! [A_107: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_107),zero_z126310315umeral) = A_107 ).

tff(fact_1969_add__0__right,axiom,
    ! [A_107: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_107),zero_zero_int) = A_107 ).

tff(fact_1970_add__0__right,axiom,
    ! [A_107: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_107),zero_zero_nat) = A_107 ).

tff(fact_1971_add__0__right,axiom,
    ! [A_107: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_107),zero_zero_real) = A_107 ).

tff(fact_1972_add__0__right,axiom,
    ! [A_107: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_107),zero_zero_complex) = A_107 ).

tff(fact_1973_add__0__right,axiom,
    ! [A_107: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_107),zero_z891286103de_int) = A_107 ).

tff(fact_1974_add__0__right,axiom,
    ! [A_107: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_107),zero_zero_rat) = A_107 ).

tff(fact_1975_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

tff(fact_1976_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

tff(fact_1977_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

tff(fact_1978_add__0,axiom,
    ! [A_106: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_106) = A_106 ).

tff(fact_1979_add__0,axiom,
    ! [A_106: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_106) = A_106 ).

tff(fact_1980_add__0,axiom,
    ! [A_106: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_106) = A_106 ).

tff(fact_1981_add__0,axiom,
    ! [A_106: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_106) = A_106 ).

tff(fact_1982_add__0,axiom,
    ! [A_106: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_106) = A_106 ).

tff(fact_1983_add__0,axiom,
    ! [A_106: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_106) = A_106 ).

tff(fact_1984_add__0,axiom,
    ! [A_106: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_106) = A_106 ).

tff(fact_1985_add__0__left,axiom,
    ! [A_105: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_105) = A_105 ).

tff(fact_1986_add__0__left,axiom,
    ! [A_105: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_105) = A_105 ).

tff(fact_1987_add__0__left,axiom,
    ! [A_105: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_105) = A_105 ).

tff(fact_1988_add__0__left,axiom,
    ! [A_105: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_105) = A_105 ).

tff(fact_1989_add__0__left,axiom,
    ! [A_105: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_105) = A_105 ).

tff(fact_1990_add__0__left,axiom,
    ! [A_105: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_105) = A_105 ).

tff(fact_1991_add__0__left,axiom,
    ! [A_105: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_105) = A_105 ).

tff(fact_1992_add__le__imp__le__left,axiom,
    ! [C_46: code_code_numeral,A_104: code_code_numeral,B_86: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),A_104)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),B_86)))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_104),B_86)) ) ).

tff(fact_1993_add__le__imp__le__left,axiom,
    ! [C_46: int,A_104: int,B_86: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),A_104)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),B_86)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_104),B_86)) ) ).

tff(fact_1994_add__le__imp__le__left,axiom,
    ! [C_46: nat,A_104: nat,B_86: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),A_104)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),B_86)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_104),B_86)) ) ).

tff(fact_1995_add__le__imp__le__left,axiom,
    ! [C_46: real,A_104: real,B_86: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),A_104)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),B_86)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_104),B_86)) ) ).

tff(fact_1996_add__le__imp__le__left,axiom,
    ! [C_46: quickcheck_code_int,A_104: quickcheck_code_int,B_86: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),A_104)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),B_86)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_104),B_86)) ) ).

tff(fact_1997_add__le__imp__le__left,axiom,
    ! [C_46: rat,A_104: rat,B_86: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),A_104)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),B_86)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_104),B_86)) ) ).

tff(fact_1998_add__le__imp__le__right,axiom,
    ! [A_103: code_code_numeral,C_45: code_code_numeral,B_85: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_103),C_45)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_85),C_45)))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_103),B_85)) ) ).

tff(fact_1999_add__le__imp__le__right,axiom,
    ! [A_103: int,C_45: int,B_85: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_103),C_45)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_85),C_45)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_103),B_85)) ) ).

tff(fact_2000_add__le__imp__le__right,axiom,
    ! [A_103: nat,C_45: nat,B_85: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_103),C_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_85),C_45)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_103),B_85)) ) ).

tff(fact_2001_add__le__imp__le__right,axiom,
    ! [A_103: real,C_45: real,B_85: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_103),C_45)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_85),C_45)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_103),B_85)) ) ).

tff(fact_2002_add__le__imp__le__right,axiom,
    ! [A_103: quickcheck_code_int,C_45: quickcheck_code_int,B_85: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_103),C_45)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_85),C_45)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_103),B_85)) ) ).

tff(fact_2003_add__le__imp__le__right,axiom,
    ! [A_103: rat,C_45: rat,B_85: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_103),C_45)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_85),C_45)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_103),B_85)) ) ).

tff(fact_2004_add__mono,axiom,
    ! [C_44: code_code_numeral,D_8: code_code_numeral,A_102: code_code_numeral,B_84: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_102),B_84))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_44),D_8))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_102),C_44)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_84),D_8))) ) ) ).

tff(fact_2005_add__mono,axiom,
    ! [C_44: int,D_8: int,A_102: int,B_84: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_102),B_84))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_44),D_8))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_102),C_44)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_84),D_8))) ) ) ).

tff(fact_2006_add__mono,axiom,
    ! [C_44: nat,D_8: nat,A_102: nat,B_84: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_102),B_84))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_44),D_8))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_102),C_44)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_84),D_8))) ) ) ).

tff(fact_2007_add__mono,axiom,
    ! [C_44: real,D_8: real,A_102: real,B_84: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_102),B_84))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_44),D_8))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_102),C_44)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_84),D_8))) ) ) ).

tff(fact_2008_add__mono,axiom,
    ! [C_44: quickcheck_code_int,D_8: quickcheck_code_int,A_102: quickcheck_code_int,B_84: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_102),B_84))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_44),D_8))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_102),C_44)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_84),D_8))) ) ) ).

tff(fact_2009_add__mono,axiom,
    ! [C_44: rat,D_8: rat,A_102: rat,B_84: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_102),B_84))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_44),D_8))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_102),C_44)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_84),D_8))) ) ) ).

tff(fact_2010_add__left__mono,axiom,
    ! [C_43: code_code_numeral,A_101: code_code_numeral,B_83: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_101),B_83))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),A_101)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),B_83))) ) ).

tff(fact_2011_add__left__mono,axiom,
    ! [C_43: int,A_101: int,B_83: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_101),B_83))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),A_101)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),B_83))) ) ).

tff(fact_2012_add__left__mono,axiom,
    ! [C_43: nat,A_101: nat,B_83: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_101),B_83))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),A_101)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),B_83))) ) ).

tff(fact_2013_add__left__mono,axiom,
    ! [C_43: real,A_101: real,B_83: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_101),B_83))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),A_101)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),B_83))) ) ).

tff(fact_2014_add__left__mono,axiom,
    ! [C_43: quickcheck_code_int,A_101: quickcheck_code_int,B_83: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_101),B_83))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),A_101)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),B_83))) ) ).

tff(fact_2015_add__left__mono,axiom,
    ! [C_43: rat,A_101: rat,B_83: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_101),B_83))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),A_101)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),B_83))) ) ).

tff(fact_2016_add__right__mono,axiom,
    ! [C_42: code_code_numeral,A_100: code_code_numeral,B_82: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_100),B_82))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_100),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_82),C_42))) ) ).

tff(fact_2017_add__right__mono,axiom,
    ! [C_42: int,A_100: int,B_82: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_100),B_82))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_100),C_42)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_82),C_42))) ) ).

tff(fact_2018_add__right__mono,axiom,
    ! [C_42: nat,A_100: nat,B_82: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_100),B_82))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_100),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_82),C_42))) ) ).

tff(fact_2019_add__right__mono,axiom,
    ! [C_42: real,A_100: real,B_82: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_100),B_82))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_100),C_42)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_82),C_42))) ) ).

tff(fact_2020_add__right__mono,axiom,
    ! [C_42: quickcheck_code_int,A_100: quickcheck_code_int,B_82: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_100),B_82))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_100),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_82),C_42))) ) ).

tff(fact_2021_add__right__mono,axiom,
    ! [C_42: rat,A_100: rat,B_82: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_100),B_82))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_100),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_82),C_42))) ) ).

tff(fact_2022_add__le__cancel__left,axiom,
    ! [C: code_code_numeral,A: code_code_numeral,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B)))
    <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)) ) ).

tff(fact_2023_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ).

tff(fact_2024_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) ) ).

tff(fact_2025_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ).

tff(fact_2026_add__le__cancel__left,axiom,
    ! [C: quickcheck_code_int,A: quickcheck_code_int,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) ) ).

tff(fact_2027_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) ) ).

tff(fact_2028_add__le__cancel__right,axiom,
    ! [A: code_code_numeral,C: code_code_numeral,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C)))
    <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)) ) ).

tff(fact_2029_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ).

tff(fact_2030_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) ) ).

tff(fact_2031_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ).

tff(fact_2032_add__le__cancel__right,axiom,
    ! [A: quickcheck_code_int,C: quickcheck_code_int,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) ) ).

tff(fact_2033_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) ) ).

tff(fact_2034_add__less__imp__less__left,axiom,
    ! [C_41: code_code_numeral,A_99: code_code_numeral,B_81: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),A_99)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),B_81)))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_99),B_81)) ) ).

tff(fact_2035_add__less__imp__less__left,axiom,
    ! [C_41: int,A_99: int,B_81: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),A_99)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),B_81)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_99),B_81)) ) ).

tff(fact_2036_add__less__imp__less__left,axiom,
    ! [C_41: nat,A_99: nat,B_81: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),A_99)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),B_81)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_99),B_81)) ) ).

tff(fact_2037_add__less__imp__less__left,axiom,
    ! [C_41: real,A_99: real,B_81: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),A_99)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),B_81)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_99),B_81)) ) ).

tff(fact_2038_add__less__imp__less__left,axiom,
    ! [C_41: quickcheck_code_int,A_99: quickcheck_code_int,B_81: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),A_99)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),B_81)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_99),B_81)) ) ).

tff(fact_2039_add__less__imp__less__left,axiom,
    ! [C_41: rat,A_99: rat,B_81: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),A_99)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),B_81)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_99),B_81)) ) ).

tff(fact_2040_add__less__imp__less__right,axiom,
    ! [A_98: code_code_numeral,C_40: code_code_numeral,B_80: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_98),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_80),C_40)))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_98),B_80)) ) ).

tff(fact_2041_add__less__imp__less__right,axiom,
    ! [A_98: int,C_40: int,B_80: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_98),C_40)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_80),C_40)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_98),B_80)) ) ).

tff(fact_2042_add__less__imp__less__right,axiom,
    ! [A_98: nat,C_40: nat,B_80: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_98),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_80),C_40)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_98),B_80)) ) ).

tff(fact_2043_add__less__imp__less__right,axiom,
    ! [A_98: real,C_40: real,B_80: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_98),C_40)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_80),C_40)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_98),B_80)) ) ).

tff(fact_2044_add__less__imp__less__right,axiom,
    ! [A_98: quickcheck_code_int,C_40: quickcheck_code_int,B_80: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_98),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_80),C_40)))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_98),B_80)) ) ).

tff(fact_2045_add__less__imp__less__right,axiom,
    ! [A_98: rat,C_40: rat,B_80: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_98),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_80),C_40)))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_98),B_80)) ) ).

tff(fact_2046_add__strict__mono,axiom,
    ! [C_39: code_code_numeral,D_7: code_code_numeral,A_97: code_code_numeral,B_79: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_97),B_79))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_39),D_7))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_97),C_39)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_79),D_7))) ) ) ).

tff(fact_2047_add__strict__mono,axiom,
    ! [C_39: int,D_7: int,A_97: int,B_79: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_97),B_79))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_39),D_7))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_97),C_39)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_79),D_7))) ) ) ).

tff(fact_2048_add__strict__mono,axiom,
    ! [C_39: nat,D_7: nat,A_97: nat,B_79: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_97),B_79))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_39),D_7))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_97),C_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_79),D_7))) ) ) ).

tff(fact_2049_add__strict__mono,axiom,
    ! [C_39: real,D_7: real,A_97: real,B_79: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_97),B_79))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_39),D_7))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_97),C_39)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_79),D_7))) ) ) ).

tff(fact_2050_add__strict__mono,axiom,
    ! [C_39: quickcheck_code_int,D_7: quickcheck_code_int,A_97: quickcheck_code_int,B_79: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_97),B_79))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_39),D_7))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_97),C_39)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_79),D_7))) ) ) ).

tff(fact_2051_add__strict__mono,axiom,
    ! [C_39: rat,D_7: rat,A_97: rat,B_79: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_97),B_79))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_39),D_7))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_97),C_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_79),D_7))) ) ) ).

tff(fact_2052_add__strict__left__mono,axiom,
    ! [C_38: code_code_numeral,A_96: code_code_numeral,B_78: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_96),B_78))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),A_96)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),B_78))) ) ).

tff(fact_2053_add__strict__left__mono,axiom,
    ! [C_38: int,A_96: int,B_78: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_78))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),A_96)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),B_78))) ) ).

tff(fact_2054_add__strict__left__mono,axiom,
    ! [C_38: nat,A_96: nat,B_78: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_78))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),A_96)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),B_78))) ) ).

tff(fact_2055_add__strict__left__mono,axiom,
    ! [C_38: real,A_96: real,B_78: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),B_78))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),A_96)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),B_78))) ) ).

tff(fact_2056_add__strict__left__mono,axiom,
    ! [C_38: quickcheck_code_int,A_96: quickcheck_code_int,B_78: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_96),B_78))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),A_96)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),B_78))) ) ).

tff(fact_2057_add__strict__left__mono,axiom,
    ! [C_38: rat,A_96: rat,B_78: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_96),B_78))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),A_96)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),B_78))) ) ).

tff(fact_2058_add__strict__right__mono,axiom,
    ! [C_37: code_code_numeral,A_95: code_code_numeral,B_77: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_95),B_77))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_95),C_37)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_77),C_37))) ) ).

tff(fact_2059_add__strict__right__mono,axiom,
    ! [C_37: int,A_95: int,B_77: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_77))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_95),C_37)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_77),C_37))) ) ).

tff(fact_2060_add__strict__right__mono,axiom,
    ! [C_37: nat,A_95: nat,B_77: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_77))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_95),C_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_77),C_37))) ) ).

tff(fact_2061_add__strict__right__mono,axiom,
    ! [C_37: real,A_95: real,B_77: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_95),B_77))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_95),C_37)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_77),C_37))) ) ).

tff(fact_2062_add__strict__right__mono,axiom,
    ! [C_37: quickcheck_code_int,A_95: quickcheck_code_int,B_77: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_95),B_77))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_95),C_37)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_77),C_37))) ) ).

tff(fact_2063_add__strict__right__mono,axiom,
    ! [C_37: rat,A_95: rat,B_77: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_95),B_77))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_95),C_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_77),C_37))) ) ).

tff(fact_2064_add__less__cancel__left,axiom,
    ! [C: code_code_numeral,A: code_code_numeral,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B)))
    <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)) ) ).

tff(fact_2065_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).

tff(fact_2066_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) ) ).

tff(fact_2067_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ).

tff(fact_2068_add__less__cancel__left,axiom,
    ! [C: quickcheck_code_int,A: quickcheck_code_int,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)) ) ).

tff(fact_2069_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) ) ).

tff(fact_2070_add__less__cancel__right,axiom,
    ! [A: code_code_numeral,C: code_code_numeral,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C)))
    <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)) ) ).

tff(fact_2071_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ).

tff(fact_2072_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) ) ).

tff(fact_2073_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ).

tff(fact_2074_add__less__cancel__right,axiom,
    ! [A: quickcheck_code_int,C: quickcheck_code_int,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C)))
    <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)) ) ).

tff(fact_2075_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) ) ).

tff(fact_2076_right__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int )
    <=> ( A = B ) ) ).

tff(fact_2077_right__minus__eq,axiom,
    ! [A: real,B: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = zero_zero_real )
    <=> ( A = B ) ) ).

tff(fact_2078_right__minus__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex )
    <=> ( A = B ) ) ).

tff(fact_2079_right__minus__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat )
    <=> ( A = B ) ) ).

tff(fact_2080_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

tff(fact_2081_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

tff(fact_2082_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex ) ) ).

tff(fact_2083_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

tff(fact_2084_diff__self,axiom,
    ! [A_94: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_94),A_94) = zero_zero_int ).

tff(fact_2085_diff__self,axiom,
    ! [A_94: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_94),A_94) = zero_zero_real ).

tff(fact_2086_diff__self,axiom,
    ! [A_94: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_94),A_94) = zero_zero_complex ).

tff(fact_2087_diff__self,axiom,
    ! [A_94: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_94),A_94) = zero_zero_rat ).

tff(fact_2088_diff__0__right,axiom,
    ! [A_93: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_93),zero_zero_int) = A_93 ).

tff(fact_2089_diff__0__right,axiom,
    ! [A_93: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_93),zero_zero_real) = A_93 ).

tff(fact_2090_diff__0__right,axiom,
    ! [A_93: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_93),zero_zero_complex) = A_93 ).

tff(fact_2091_diff__0__right,axiom,
    ! [A_93: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_93),zero_zero_rat) = A_93 ).

tff(fact_2092_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D)) ) ) ).

tff(fact_2093_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D)) ) ) ).

tff(fact_2094_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D)) ) ) ).

tff(fact_2095_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D)) ) ) ).

tff(fact_2096_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D)) ) ) ).

tff(fact_2097_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D)) ) ) ).

tff(fact_2098_mult_Ocomm__neutral,axiom,
    ! [A_92: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_92),one_one_int) = A_92 ).

tff(fact_2099_mult_Ocomm__neutral,axiom,
    ! [A_92: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_92),one_one_nat) = A_92 ).

tff(fact_2100_mult_Ocomm__neutral,axiom,
    ! [A_92: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_92),one_one_real) = A_92 ).

tff(fact_2101_mult_Ocomm__neutral,axiom,
    ! [A_92: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_92),one_on1645066479umeral) = A_92 ).

tff(fact_2102_mult_Ocomm__neutral,axiom,
    ! [A_92: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_92),one_one_complex) = A_92 ).

tff(fact_2103_mult_Ocomm__neutral,axiom,
    ! [A_92: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_92),one_on1684967323de_int) = A_92 ).

tff(fact_2104_mult_Ocomm__neutral,axiom,
    ! [A_92: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_92),one_one_rat) = A_92 ).

tff(fact_2105_mult__1__right,axiom,
    ! [A_91: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_91),one_one_int) = A_91 ).

tff(fact_2106_mult__1__right,axiom,
    ! [A_91: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_91),one_one_nat) = A_91 ).

tff(fact_2107_mult__1__right,axiom,
    ! [A_91: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_91),one_one_real) = A_91 ).

tff(fact_2108_mult__1__right,axiom,
    ! [A_91: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_91),one_on1645066479umeral) = A_91 ).

tff(fact_2109_mult__1__right,axiom,
    ! [A_91: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_91),one_one_complex) = A_91 ).

tff(fact_2110_mult__1__right,axiom,
    ! [A_91: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_91),one_on1684967323de_int) = A_91 ).

tff(fact_2111_mult__1__right,axiom,
    ! [A_91: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_91),one_one_rat) = A_91 ).

tff(fact_2112_mult__1,axiom,
    ! [A_90: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_90) = A_90 ).

tff(fact_2113_mult__1,axiom,
    ! [A_90: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_90) = A_90 ).

tff(fact_2114_mult__1,axiom,
    ! [A_90: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_90) = A_90 ).

tff(fact_2115_mult__1,axiom,
    ! [A_90: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_90) = A_90 ).

tff(fact_2116_mult__1,axiom,
    ! [A_90: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_90) = A_90 ).

tff(fact_2117_mult__1,axiom,
    ! [A_90: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_90) = A_90 ).

tff(fact_2118_mult__1,axiom,
    ! [A_90: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_90) = A_90 ).

tff(fact_2119_mult__1__left,axiom,
    ! [A_89: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_89) = A_89 ).

tff(fact_2120_mult__1__left,axiom,
    ! [A_89: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_89) = A_89 ).

tff(fact_2121_mult__1__left,axiom,
    ! [A_89: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_89) = A_89 ).

tff(fact_2122_mult__1__left,axiom,
    ! [A_89: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_89) = A_89 ).

tff(fact_2123_mult__1__left,axiom,
    ! [A_89: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_89) = A_89 ).

tff(fact_2124_mult__1__left,axiom,
    ! [A_89: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_89) = A_89 ).

tff(fact_2125_mult__1__left,axiom,
    ! [A_89: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_89) = A_89 ).

tff(fact_2126_inv__less__p__minus__1,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(inv(P_3),A_3)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) ) ) ) ).

tff(fact_2127_inv__g__1,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(inv(P_3),A_3))) ) ) ) ).

tff(fact_2128_inv__not__p__minus__1,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( hAPP_int_int(inv(P_3),A_3) != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) ) ) ) ) ).

tff(fact_2129_inv__not__1,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( hAPP_int_int(inv(P_3),A_3) != one_one_int ) ) ) ) ).

tff(fact_2130_inv__distinct,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( A_3 != hAPP_int_int(inv(P_3),A_3) ) ) ) ) ).

tff(fact_2131_add__diff__cancel,axiom,
    ! [A_88: int,B_76: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_88),B_76)),B_76) = A_88 ).

tff(fact_2132_add__diff__cancel,axiom,
    ! [A_88: real,B_76: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_88),B_76)),B_76) = A_88 ).

tff(fact_2133_add__diff__cancel,axiom,
    ! [A_88: complex,B_76: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_88),B_76)),B_76) = A_88 ).

tff(fact_2134_add__diff__cancel,axiom,
    ! [A_88: rat,B_76: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_88),B_76)),B_76) = A_88 ).

tff(fact_2135_diff__add__cancel,axiom,
    ! [A_87: int,B_75: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_87),B_75)),B_75) = A_87 ).

tff(fact_2136_diff__add__cancel,axiom,
    ! [A_87: real,B_75: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_87),B_75)),B_75) = A_87 ).

tff(fact_2137_diff__add__cancel,axiom,
    ! [A_87: complex,B_75: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_87),B_75)),B_75) = A_87 ).

tff(fact_2138_diff__add__cancel,axiom,
    ! [A_87: rat,B_75: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_87),B_75)),B_75) = A_87 ).

tff(fact_2139_inv__not__0,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))
         => ( hAPP_int_int(inv(P_3),A_3) != zero_zero_int ) ) ) ) ).

tff(fact_2140_inv__is__inv,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(inv(P_3),A_3))),one_one_int),P_3)) ) ) ) ).

tff(fact_2141_add__nonpos__nonpos,axiom,
    ! [B_74: code_code_numeral,A_86: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_86),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_74),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_86),B_74)),zero_z126310315umeral)) ) ) ).

tff(fact_2142_add__nonpos__nonpos,axiom,
    ! [B_74: int,A_86: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_86),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_74),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_86),B_74)),zero_zero_int)) ) ) ).

tff(fact_2143_add__nonpos__nonpos,axiom,
    ! [B_74: nat,A_86: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_86),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_74),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_86),B_74)),zero_zero_nat)) ) ) ).

tff(fact_2144_add__nonpos__nonpos,axiom,
    ! [B_74: real,A_86: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_86),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_74),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_86),B_74)),zero_zero_real)) ) ) ).

tff(fact_2145_add__nonpos__nonpos,axiom,
    ! [B_74: quickcheck_code_int,A_86: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_86),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_74),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_86),B_74)),zero_z891286103de_int)) ) ) ).

tff(fact_2146_add__nonpos__nonpos,axiom,
    ! [B_74: rat,A_86: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_86),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_74),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_86),B_74)),zero_zero_rat)) ) ) ).

tff(fact_2147_add__increasing2,axiom,
    ! [B_73: code_code_numeral,A_85: code_code_numeral,C_36: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_36))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),A_85))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_85),C_36))) ) ) ).

tff(fact_2148_add__increasing2,axiom,
    ! [B_73: int,A_85: int,C_36: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_36))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),A_85))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_85),C_36))) ) ) ).

tff(fact_2149_add__increasing2,axiom,
    ! [B_73: nat,A_85: nat,C_36: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_36))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),A_85))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_85),C_36))) ) ) ).

tff(fact_2150_add__increasing2,axiom,
    ! [B_73: real,A_85: real,C_36: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_36))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),A_85))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_85),C_36))) ) ) ).

tff(fact_2151_add__increasing2,axiom,
    ! [B_73: quickcheck_code_int,A_85: quickcheck_code_int,C_36: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_36))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),A_85))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_85),C_36))) ) ) ).

tff(fact_2152_add__increasing2,axiom,
    ! [B_73: rat,A_85: rat,C_36: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_36))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),A_85))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_85),C_36))) ) ) ).

tff(fact_2153_add__increasing,axiom,
    ! [B_72: code_code_numeral,C_35: code_code_numeral,A_84: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_84))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),C_35))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_84),C_35))) ) ) ).

tff(fact_2154_add__increasing,axiom,
    ! [B_72: int,C_35: int,A_84: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_84))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),C_35))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_84),C_35))) ) ) ).

tff(fact_2155_add__increasing,axiom,
    ! [B_72: nat,C_35: nat,A_84: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_84))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),C_35))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_84),C_35))) ) ) ).

tff(fact_2156_add__increasing,axiom,
    ! [B_72: real,C_35: real,A_84: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_84))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),C_35))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_84),C_35))) ) ) ).

tff(fact_2157_add__increasing,axiom,
    ! [B_72: quickcheck_code_int,C_35: quickcheck_code_int,A_84: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_84))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),C_35))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_84),C_35))) ) ) ).

tff(fact_2158_add__increasing,axiom,
    ! [B_72: rat,C_35: rat,A_84: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_84))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),C_35))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_84),C_35))) ) ) ).

tff(fact_2159_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: code_code_numeral,X_1: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_1))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_2))
       => ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_1),Y_2) = zero_z126310315umeral )
        <=> ( ( X_1 = zero_z126310315umeral )
            & ( Y_2 = zero_z126310315umeral ) ) ) ) ) ).

tff(fact_2160_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2) = zero_zero_int )
        <=> ( ( X_1 = zero_zero_int )
            & ( Y_2 = zero_zero_int ) ) ) ) ) ).

tff(fact_2161_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_2))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2) = zero_zero_nat )
        <=> ( ( X_1 = zero_zero_nat )
            & ( Y_2 = zero_zero_nat ) ) ) ) ) ).

tff(fact_2162_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2))
       => ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real )
        <=> ( ( X_1 = zero_zero_real )
            & ( Y_2 = zero_zero_real ) ) ) ) ) ).

tff(fact_2163_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: quickcheck_code_int,X_1: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_1))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_2))
       => ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_1),Y_2) = zero_z891286103de_int )
        <=> ( ( X_1 = zero_z891286103de_int )
            & ( Y_2 = zero_z891286103de_int ) ) ) ) ) ).

tff(fact_2164_add__nonneg__eq__0__iff,axiom,
    ! [Y_2: rat,X_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_1))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_2))
       => ( ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Y_2) = zero_zero_rat )
        <=> ( ( X_1 = zero_zero_rat )
            & ( Y_2 = zero_zero_rat ) ) ) ) ) ).

tff(fact_2165_add__nonneg__nonneg,axiom,
    ! [B_71: code_code_numeral,A_83: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_83))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_71))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_83),B_71))) ) ) ).

tff(fact_2166_add__nonneg__nonneg,axiom,
    ! [B_71: int,A_83: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_83))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_71))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_83),B_71))) ) ) ).

tff(fact_2167_add__nonneg__nonneg,axiom,
    ! [B_71: nat,A_83: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_83))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_71))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_83),B_71))) ) ) ).

tff(fact_2168_add__nonneg__nonneg,axiom,
    ! [B_71: real,A_83: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_83))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_71))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_83),B_71))) ) ) ).

tff(fact_2169_add__nonneg__nonneg,axiom,
    ! [B_71: quickcheck_code_int,A_83: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_83))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_71))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_83),B_71))) ) ) ).

tff(fact_2170_add__nonneg__nonneg,axiom,
    ! [B_71: rat,A_83: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_83))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_71))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_83),B_71))) ) ) ).

tff(fact_2171_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

tff(fact_2172_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

tff(fact_2173_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

tff(fact_2174_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

tff(fact_2175_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

tff(fact_2176_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

tff(fact_2177_add__neg__neg,axiom,
    ! [B_70: code_code_numeral,A_82: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_82),zero_z126310315umeral))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_70),zero_z126310315umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_82),B_70)),zero_z126310315umeral)) ) ) ).

tff(fact_2178_add__neg__neg,axiom,
    ! [B_70: int,A_82: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_82),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_70),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_82),B_70)),zero_zero_int)) ) ) ).

tff(fact_2179_add__neg__neg,axiom,
    ! [B_70: nat,A_82: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_82),zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_70),zero_zero_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_82),B_70)),zero_zero_nat)) ) ) ).

tff(fact_2180_add__neg__neg,axiom,
    ! [B_70: real,A_82: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_82),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_70),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_82),B_70)),zero_zero_real)) ) ) ).

tff(fact_2181_add__neg__neg,axiom,
    ! [B_70: quickcheck_code_int,A_82: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_82),zero_z891286103de_int))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_70),zero_z891286103de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_82),B_70)),zero_z891286103de_int)) ) ) ).

tff(fact_2182_add__neg__neg,axiom,
    ! [B_70: rat,A_82: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_82),zero_zero_rat))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_70),zero_zero_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_82),B_70)),zero_zero_rat)) ) ) ).

tff(fact_2183_add__pos__pos,axiom,
    ! [B_69: code_code_numeral,A_81: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_81))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_69))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_81),B_69))) ) ) ).

tff(fact_2184_add__pos__pos,axiom,
    ! [B_69: int,A_81: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_81))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_69))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_81),B_69))) ) ) ).

tff(fact_2185_add__pos__pos,axiom,
    ! [B_69: nat,A_81: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_81))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_69))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),B_69))) ) ) ).

tff(fact_2186_add__pos__pos,axiom,
    ! [B_69: real,A_81: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_81))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_69))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_81),B_69))) ) ) ).

tff(fact_2187_add__pos__pos,axiom,
    ! [B_69: quickcheck_code_int,A_81: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_81))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_69))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_81),B_69))) ) ) ).

tff(fact_2188_add__pos__pos,axiom,
    ! [B_69: rat,A_81: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_81))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_69))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_81),B_69))) ) ) ).

tff(fact_2189_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ).

tff(fact_2190_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)) ) ).

tff(fact_2191_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)) ) ).

tff(fact_2192_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) ) ).

tff(fact_2193_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) ) ).

tff(fact_2194_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A)) ) ).

tff(fact_2195_add__le__less__mono,axiom,
    ! [C_34: code_code_numeral,D_6: code_code_numeral,A_80: code_code_numeral,B_68: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_80),B_68))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_34),D_6))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_80),C_34)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_68),D_6))) ) ) ).

tff(fact_2196_add__le__less__mono,axiom,
    ! [C_34: int,D_6: int,A_80: int,B_68: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_80),B_68))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),D_6))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_80),C_34)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_68),D_6))) ) ) ).

tff(fact_2197_add__le__less__mono,axiom,
    ! [C_34: nat,D_6: nat,A_80: nat,B_68: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_80),B_68))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),D_6))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_80),C_34)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_68),D_6))) ) ) ).

tff(fact_2198_add__le__less__mono,axiom,
    ! [C_34: real,D_6: real,A_80: real,B_68: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_80),B_68))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_34),D_6))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_80),C_34)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_68),D_6))) ) ) ).

tff(fact_2199_add__le__less__mono,axiom,
    ! [C_34: quickcheck_code_int,D_6: quickcheck_code_int,A_80: quickcheck_code_int,B_68: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_80),B_68))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_34),D_6))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_80),C_34)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_68),D_6))) ) ) ).

tff(fact_2200_add__le__less__mono,axiom,
    ! [C_34: rat,D_6: rat,A_80: rat,B_68: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_80),B_68))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_34),D_6))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_80),C_34)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_68),D_6))) ) ) ).

tff(fact_2201_add__less__le__mono,axiom,
    ! [C_33: code_code_numeral,D_5: code_code_numeral,A_79: code_code_numeral,B_67: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_79),B_67))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_33),D_5))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_79),C_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_67),D_5))) ) ) ).

tff(fact_2202_add__less__le__mono,axiom,
    ! [C_33: int,D_5: int,A_79: int,B_67: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_67))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_33),D_5))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_79),C_33)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_67),D_5))) ) ) ).

tff(fact_2203_add__less__le__mono,axiom,
    ! [C_33: nat,D_5: nat,A_79: nat,B_67: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_67))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_33),D_5))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),C_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_67),D_5))) ) ) ).

tff(fact_2204_add__less__le__mono,axiom,
    ! [C_33: real,D_5: real,A_79: real,B_67: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_67))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_33),D_5))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_79),C_33)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_67),D_5))) ) ) ).

tff(fact_2205_add__less__le__mono,axiom,
    ! [C_33: quickcheck_code_int,D_5: quickcheck_code_int,A_79: quickcheck_code_int,B_67: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_79),B_67))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_33),D_5))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_79),C_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_67),D_5))) ) ) ).

tff(fact_2206_add__less__le__mono,axiom,
    ! [C_33: rat,D_5: rat,A_79: rat,B_67: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_79),B_67))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_33),D_5))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_79),C_33)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_67),D_5))) ) ) ).

tff(fact_2207_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

tff(fact_2208_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

tff(fact_2209_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).

tff(fact_2210_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

tff(fact_2211_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

tff(fact_2212_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) ) ).

tff(fact_2213_zfact_Osimps,axiom,
    ! [N: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
       => ( hAPP_int_int(zfact,N) = one_one_int ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
       => ( hAPP_int_int(zfact,N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))) ) ) ) ).

tff(fact_2214_wset__mem__inv__mem,axiom,
    ! [B: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ) ) ).

tff(fact_2215_wset__inv__mem__mem,axiom,
    ! [B: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
             => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5)))
               => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ) ) ) ) ).

tff(fact_2216_MultInvPair__distinct,axiom,
    ! [J_2: int,A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
           => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
             => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ) ).

tff(fact_2217_int__le__induct,axiom,
    ! [P_1: fun_int_bool,I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))
     => ( hBOOL(hAPP_int_bool(P_1,K_1))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_2218_d22set__induct__old,axiom,
    ! [X_1: int,P_1: fun_int_bool] :
      ( ! [A_2: int] :
          ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),one_one_int))) )
         => hBOOL(hAPP_int_bool(P_1,A_2)) )
     => hBOOL(hAPP_int_bool(P_1,X_1)) ) ).

tff(fact_2219_int__less__induct,axiom,
    ! [P_1: fun_int_bool,I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1))
     => ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int)))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_2220_int__ge__induct,axiom,
    ! [P_1: fun_int_bool,K_1: int,I_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))
     => ( hBOOL(hAPP_int_bool(P_1,K_1))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_2221_aux______1,axiom,
    ! [J_2: int,A_3: int,P_3: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)) ) ).

tff(fact_2222_aux______3,axiom,
    ! [J_2: int,K_2: int,A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3)) ) ).

tff(fact_2223_wset__mem__mem,axiom,
    ! [P_5: int,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),hAPP_i1948725293t_bool(wset(A),P_5))) ) ).

tff(fact_2224_wset__subset,axiom,
    ! [B: int,P_5: int,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5)))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ).

tff(fact_2225_wset__g__1,axiom,
    ! [B: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) ) ) ) ).

tff(fact_2226_wset__less,axiom,
    ! [B: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) ) ) ) ).

tff(fact_2227_wset__mem__imp__or,axiom,
    ! [B: int,P_5: int,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))
         => ( ( B = A )
            | ( B = hAPP_int_int(inv(P_5),A) ) ) ) ) ) ).

tff(fact_2228_MultInv__prop1,axiom,
    ! [X: int,Y: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3)) ) ) ).

tff(fact_2229_wset__mem,axiom,
    ! [B: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) ) ) ) ) ).

tff(fact_2230_MultInv__zcong__prop1,axiom,
    ! [A_3: int,J_2: int,K_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) ) ) ).

tff(fact_2231_MultInv__prop3,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),zero_zero_int),P_3)) ) ) ) ).

tff(fact_2232_MultInv__prop4,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),X),P_3)) ) ) ) ).

tff(fact_2233_MultInv__prop5,axiom,
    ! [Y: int,X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3)) ) ) ) ) ) ).

tff(fact_2234_Int2_Oaux____1,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),P_3)) ) ) ) ).

tff(fact_2235_Int2_Oaux____2,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),X),P_3)) ) ) ) ).

tff(fact_2236_MultInv__zcong__prop3,axiom,
    ! [J_2: int,K_2: int,A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
           => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3)) ) ) ) ) ) ) ).

tff(fact_2237_MultInv__zcong__prop2,axiom,
    ! [A_3: int,J_2: int,K_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ) ).

tff(fact_2238_aux______2,axiom,
    ! [J_2: int,A_3: int,K_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3)) ) ) ) ) ).

tff(fact_2239_aux______4,axiom,
    ! [K_2: int,A_3: int,J_2: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)) ) ) ) ) ).

tff(fact_2240_MultInv__prop2,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),one_one_int),P_3)) ) ) ) ).

tff(fact_2241_MultInv__prop2a,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),X)),X)),one_one_int),P_3)) ) ) ) ).

tff(fact_2242_int__gr__induct,axiom,
    ! [P_1: fun_int_bool,K_1: int,I_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1))
     => ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int)))
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_2243_mono__nat__linear__lb,axiom,
    ! [Ma: nat,K_1: nat,F: fun_nat_nat] :
      ( ! [M_1: nat,N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_2))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,Ma)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)))) ) ).

tff(fact_2244_d22set__eq__wset,axiom,
    ! [P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hAPP_i1948725293t_bool(d22set,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5) ) ) ).

tff(fact_2245_ex__least__nat__le,axiom,
    ! [N_1: nat,P_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(P_1,N_1))
       => ? [K: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1))
            & ! [I_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K))
               => ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
            & hBOOL(hAPP_nat_bool(P_1,K)) ) ) ) ).

tff(fact_2246_less__imp__add__positive,axiom,
    ! [I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ? [K: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
          & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K) = J_2 ) ) ) ).

tff(fact_2247_minf_I10_J,axiom,
    ! [D: code_code_numeral,Sa: code_code_numeral] :
    ? [Z: code_code_numeral] :
    ! [X_2: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_2),Z))
     => ( ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))
      <=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) ) ) ).

tff(fact_2248_minf_I10_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Z))
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))
      <=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) ) ) ).

tff(fact_2249_minf_I10_J,axiom,
    ! [D: quickcheck_code_int,Sa: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_2),Z))
     => ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))
      <=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) ) ) ).

tff(fact_2250_minf_I10_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Z))
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))
      <=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) ) ) ).

tff(fact_2251_minf_I10_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))
      <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) ) ) ).

tff(fact_2252_minf_I10_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Z))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))
      <=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) ) ) ).

tff(fact_2253_pinf_I10_J,axiom,
    ! [D: code_code_numeral,Sa: code_code_numeral] :
    ? [Z: code_code_numeral] :
    ! [X_2: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X_2))
     => ( ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))
      <=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) ) ) ).

tff(fact_2254_pinf_I10_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X_2))
     => ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))
      <=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) ) ) ).

tff(fact_2255_pinf_I10_J,axiom,
    ! [D: quickcheck_code_int,Sa: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X_2))
     => ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))
      <=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) ) ) ).

tff(fact_2256_pinf_I10_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X_2))
     => ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))
      <=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) ) ) ).

tff(fact_2257_pinf_I10_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))
      <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) ) ) ).

tff(fact_2258_pinf_I10_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_2))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))
      <=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) ) ) ).

tff(fact_2259_minf_I9_J,axiom,
    ! [D: code_code_numeral,Sa: code_code_numeral] :
    ? [Z: code_code_numeral] :
    ! [X_2: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_2),Z))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))
      <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) ) ) ).

tff(fact_2260_minf_I9_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Z))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) ) ) ).

tff(fact_2261_minf_I9_J,axiom,
    ! [D: quickcheck_code_int,Sa: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_2),Z))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))
      <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) ) ) ).

tff(fact_2262_minf_I9_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Z))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) ) ) ).

tff(fact_2263_minf_I9_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) ) ) ).

tff(fact_2264_minf_I9_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Z))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) ) ) ).

tff(fact_2265_d22set__le__swap,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) ) ).

tff(fact_2266_d22set__le,axiom,
    ! [B: int,A: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ).

tff(fact_2267_d22set__g__1,axiom,
    ! [B: int,A: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) ) ).

tff(fact_2268_d22set__mem,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) ) ) ).

tff(fact_2269_pinf_I9_J,axiom,
    ! [D: code_code_numeral,Sa: code_code_numeral] :
    ? [Z: code_code_numeral] :
    ! [X_2: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X_2))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))
      <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) ) ) ).

tff(fact_2270_pinf_I9_J,axiom,
    ! [D: real,Sa: real] :
    ? [Z: real] :
    ! [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) ) ) ).

tff(fact_2271_pinf_I9_J,axiom,
    ! [D: quickcheck_code_int,Sa: quickcheck_code_int] :
    ? [Z: quickcheck_code_int] :
    ! [X_2: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X_2))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))
      <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) ) ) ).

tff(fact_2272_pinf_I9_J,axiom,
    ! [D: rat,Sa: rat] :
    ? [Z: rat] :
    ! [X_2: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X_2))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))
      <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) ) ) ).

tff(fact_2273_pinf_I9_J,axiom,
    ! [D: int,Sa: int] :
    ? [Z: int] :
    ! [X_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) ) ) ).

tff(fact_2274_pinf_I9_J,axiom,
    ! [D: nat,Sa: nat] :
    ? [Z: nat] :
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) ) ) ).

tff(fact_2275_pow__divides__eq__int,axiom,
    ! [A: int,B: int,N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),N_1)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) ) ) ).

tff(fact_2276_pow__divides__pow__int,axiom,
    ! [A_3: int,N: nat,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)))
     => ( ( N != zero_zero_nat )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2)) ) ) ).

tff(fact_2277_pow__divides__eq__nat,axiom,
    ! [A: nat,B: nat,N_1: nat] :
      ( ( N_1 != zero_zero_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) ) ) ).

tff(fact_2278_divides__le,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
        | ( N = zero_zero_nat ) ) ) ).

tff(fact_2279_mult__left__cancel,axiom,
    ! [N: nat,M: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M) )
       => ( N = M ) ) ) ).

tff(fact_2280_dvd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

tff(fact_2281_mult_Oprod__diff__prod,axiom,
    ! [X_18: real,Y_15: real,A_78: real,B_66: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_18),Y_15)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),B_66)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_18),A_78)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_15),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_18),A_78)),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_15),B_66))) ).

tff(fact_2282_mult_Oprod__diff__prod,axiom,
    ! [X_18: complex,Y_15: complex,A_78: complex,B_66: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_18),Y_15)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_78),B_66)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_18),A_78)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_15),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_18),A_78)),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_78),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_15),B_66))) ).

tff(fact_2283_mult_Ozero__left,axiom,
    ! [B_65: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),B_65) = zero_zero_real ).

tff(fact_2284_mult_Ozero__left,axiom,
    ! [B_65: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),B_65) = zero_zero_complex ).

tff(fact_2285_mult__left_Ozero,axiom,
    ! [Y_14: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),Y_14) = zero_zero_real ).

tff(fact_2286_mult__left_Ozero,axiom,
    ! [Y_14: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),Y_14) = zero_zero_complex ).

tff(fact_2287_mult_Ozero__right,axiom,
    ! [A_77: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),zero_zero_real) = zero_zero_real ).

tff(fact_2288_mult_Ozero__right,axiom,
    ! [A_77: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_77),zero_zero_complex) = zero_zero_complex ).

tff(fact_2289_mult__right_Ozero,axiom,
    ! [X_17: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_17),zero_zero_real) = zero_zero_real ).

tff(fact_2290_mult__right_Ozero,axiom,
    ! [X_17: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_17),zero_zero_complex) = zero_zero_complex ).

tff(fact_2291_mult_Oadd__right,axiom,
    ! [A_76: real,B_64: real,B_63: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_64),B_63)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),B_64)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),B_63)) ).

tff(fact_2292_mult_Oadd__right,axiom,
    ! [A_76: complex,B_64: complex,B_63: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_76),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_64),B_63)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_76),B_64)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_76),B_63)) ).

tff(fact_2293_mult__right_Oadd,axiom,
    ! [Xa_4: real,X_16: real,Y_13: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_16),Y_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),X_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),Y_13)) ).

tff(fact_2294_mult__right_Oadd,axiom,
    ! [Xa_4: complex,X_16: complex,Y_13: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_16),Y_13)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),X_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),Y_13)) ).

tff(fact_2295_mult_Oadd__left,axiom,
    ! [A_75: real,A_74: real,B_62: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_75),A_74)),B_62) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_75),B_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_74),B_62)) ).

tff(fact_2296_mult_Oadd__left,axiom,
    ! [A_75: complex,A_74: complex,B_62: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_75),A_74)),B_62) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_75),B_62)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_74),B_62)) ).

tff(fact_2297_mult__left_Oadd,axiom,
    ! [X_15: real,Y_12: real,Ya_2: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_15),Y_12)),Ya_2) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_15),Ya_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_12),Ya_2)) ).

tff(fact_2298_mult__left_Oadd,axiom,
    ! [X_15: complex,Y_12: complex,Ya_2: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_15),Y_12)),Ya_2) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_15),Ya_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_12),Ya_2)) ).

tff(fact_2299_mult__left_Odiff,axiom,
    ! [X_14: real,Y_11: real,Ya_1: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_14),Y_11)),Ya_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_14),Ya_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_11),Ya_1)) ).

tff(fact_2300_mult__left_Odiff,axiom,
    ! [X_14: complex,Y_11: complex,Ya_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_14),Y_11)),Ya_1) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_14),Ya_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_11),Ya_1)) ).

tff(fact_2301_mult_Odiff__left,axiom,
    ! [A_73: real,A_72: real,B_61: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_73),A_72)),B_61) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_73),B_61)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_72),B_61)) ).

tff(fact_2302_mult_Odiff__left,axiom,
    ! [A_73: complex,A_72: complex,B_61: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_73),A_72)),B_61) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_73),B_61)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_72),B_61)) ).

tff(fact_2303_mult__right_Odiff,axiom,
    ! [Xa_3: real,X_13: real,Y_10: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_13),Y_10)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),X_13)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),Y_10)) ).

tff(fact_2304_mult__right_Odiff,axiom,
    ! [Xa_3: complex,X_13: complex,Y_10: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_13),Y_10)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),X_13)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),Y_10)) ).

tff(fact_2305_mult_Odiff__right,axiom,
    ! [A_71: real,B_60: real,B_59: real] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_71),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_60),B_59)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_71),B_60)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_71),B_59)) ).

tff(fact_2306_mult_Odiff__right,axiom,
    ! [A_71: complex,B_60: complex,B_59: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_71),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_60),B_59)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_71),B_60)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_71),B_59)) ).

tff(fact_2307_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
    ! [A_3: nat] :
      ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)) ) ).

tff(fact_2308_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
    ! [A_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)) ).

tff(fact_2309_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)) ) ) ).

tff(fact_2310_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
    ! [A: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

tff(fact_2311_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
    ! [A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3))
     => ( A_3 = zero_zero_nat ) ) ).

tff(fact_2312_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
    ! [A_3: nat] :
      ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)) ) ).

tff(fact_2313_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
    ! [A_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)) ).

tff(fact_2314_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
    ! [A: nat] :
      ( ( A != one_one_nat )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

tff(fact_2315_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
    ! [A: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat))
    <=> ( A = one_one_nat ) ) ).

tff(fact_2316_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
    ! [A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat))
     => ( A_3 = one_one_nat ) ) ).

tff(fact_2317_SR__def,axiom,
    ! [P_5: int] : hAPP_i1948725293t_bool(sr,P_5) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) ).

tff(fact_2318_zmod__number__of__Bit1,axiom,
    ! [V_2: int,W: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)))),one_one_int) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)))),one_one_int) ) ) ) ).

tff(fact_2319_neg__zmod__mult__2,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))),one_one_int) ) ) ).

tff(fact_2320_StandardRes__prop4,axiom,
    ! [X: int,Y: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(standardRes(M),X)),hAPP_int_int(standardRes(M),Y))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M)) ) ).

tff(fact_2321_negDivAlg__eqn__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))))
         => ( negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W))))
         => ( negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),negDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) ) ) ) ) ).

tff(fact_2322_zOddI,axiom,
    ! [X_1: int,K_1: int] :
      ( ( X_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1)),one_one_int) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) ) ).

tff(fact_2323_pos__zmod__mult__2,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(B_2),A_3))) ) ) ).

tff(fact_2324_pos__zdiv__mult__2,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),A_3) ) ) ).

tff(fact_2325_mod__mod__trivial,axiom,
    ! [A_70: int,B_58: int] : hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_70),B_58)),B_58) = hAPP_int_int(div_mod_int(A_70),B_58) ).

tff(fact_2326_mod__mod__trivial,axiom,
    ! [A_70: nat,B_58: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_70),B_58)),B_58) = hAPP_nat_nat(div_mod_nat(A_70),B_58) ).

tff(fact_2327_mod__mod__trivial,axiom,
    ! [A_70: code_code_numeral,B_58: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_70),B_58)),B_58) = hAPP_C498520661umeral(div_mo1740067990umeral(A_70),B_58) ).

tff(fact_2328_mod__mod__trivial,axiom,
    ! [A_70: quickcheck_code_int,B_58: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_70),B_58)),B_58) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_70),B_58) ).

tff(fact_2329_zmod__zdiv__trivial,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2) = zero_zero_int ).

tff(fact_2330_DIVISION__BY__ZERO,axiom,
    ! [A_3: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),zero_zero_int) = zero_zero_int )
      & ( hAPP_int_int(div_mod_int(A_3),zero_zero_int) = A_3 ) ) ).

tff(fact_2331_mod__div__trivial,axiom,
    ! [A_69: int,B_57: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_69),B_57)),B_57) = zero_zero_int ).

tff(fact_2332_mod__div__trivial,axiom,
    ! [A_69: nat,B_57: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(div_mod_nat(A_69),B_57)),B_57) = zero_zero_nat ).

tff(fact_2333_mod__div__trivial,axiom,
    ! [A_69: code_code_numeral,B_57: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_69),B_57)),B_57) = zero_z126310315umeral ).

tff(fact_2334_mod__div__trivial,axiom,
    ! [A_69: quickcheck_code_int,B_57: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_69),B_57)),B_57) = zero_z891286103de_int ).

tff(fact_2335_StandardRes__SR__prop,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sr,P_5)))
     => ( hAPP_int_int(standardRes(P_5),X_1) = X_1 ) ) ).

tff(fact_2336_StandardRes__def,axiom,
    ! [M: int,X: int] : hAPP_int_int(standardRes(M),X) = hAPP_int_int(div_mod_int(X),M) ).

tff(fact_2337_zdiv__zadd1__eq,axiom,
    ! [A_3: int,B_2: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_3),C_1)),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) ).

tff(fact_2338_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_68: int,B_56: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_68),B_56)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_68),B_56)),B_56)) = A_68 ).

tff(fact_2339_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_68: nat,B_56: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_68),B_56)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_68),B_56)),B_56)) = A_68 ).

tff(fact_2340_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_68: code_code_numeral,B_56: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_68),B_56)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_68),B_56)),B_56)) = A_68 ).

tff(fact_2341_semiring__div__class_Omod__div__equality_H,axiom,
    ! [A_68: quickcheck_code_int,B_56: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_68),B_56)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_68),B_56)),B_56)) = A_68 ).

tff(fact_2342_mod__div__equality2,axiom,
    ! [B_55: int,A_67: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_55),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_67),B_55))),hAPP_int_int(div_mod_int(A_67),B_55)) = A_67 ).

tff(fact_2343_mod__div__equality2,axiom,
    ! [B_55: nat,A_67: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_55),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_67),B_55))),hAPP_nat_nat(div_mod_nat(A_67),B_55)) = A_67 ).

tff(fact_2344_mod__div__equality2,axiom,
    ! [B_55: code_code_numeral,A_67: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_55),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_67),B_55))),hAPP_C498520661umeral(div_mo1740067990umeral(A_67),B_55)) = A_67 ).

tff(fact_2345_mod__div__equality2,axiom,
    ! [B_55: quickcheck_code_int,A_67: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_55),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_67),B_55))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_67),B_55)) = A_67 ).

tff(fact_2346_mod__div__equality,axiom,
    ! [A_66: int,B_54: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_66),B_54)),B_54)),hAPP_int_int(div_mod_int(A_66),B_54)) = A_66 ).

tff(fact_2347_mod__div__equality,axiom,
    ! [A_66: nat,B_54: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_66),B_54)),B_54)),hAPP_nat_nat(div_mod_nat(A_66),B_54)) = A_66 ).

tff(fact_2348_mod__div__equality,axiom,
    ! [A_66: code_code_numeral,B_54: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_66),B_54)),B_54)),hAPP_C498520661umeral(div_mo1740067990umeral(A_66),B_54)) = A_66 ).

tff(fact_2349_mod__div__equality,axiom,
    ! [A_66: quickcheck_code_int,B_54: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_66),B_54)),B_54)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_66),B_54)) = A_66 ).

tff(fact_2350_div__mod__equality2,axiom,
    ! [B_53: int,A_65: int,C_32: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_53),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_65),B_53))),hAPP_int_int(div_mod_int(A_65),B_53))),C_32) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_65),C_32) ).

tff(fact_2351_div__mod__equality2,axiom,
    ! [B_53: nat,A_65: nat,C_32: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_65),B_53))),hAPP_nat_nat(div_mod_nat(A_65),B_53))),C_32) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_65),C_32) ).

tff(fact_2352_div__mod__equality2,axiom,
    ! [B_53: code_code_numeral,A_65: code_code_numeral,C_32: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_53),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_65),B_53))),hAPP_C498520661umeral(div_mo1740067990umeral(A_65),B_53))),C_32) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_65),C_32) ).

tff(fact_2353_div__mod__equality2,axiom,
    ! [B_53: quickcheck_code_int,A_65: quickcheck_code_int,C_32: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_53),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_65),B_53))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_65),B_53))),C_32) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_65),C_32) ).

tff(fact_2354_div__mod__equality,axiom,
    ! [A_64: int,B_52: int,C_31: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_64),B_52)),B_52)),hAPP_int_int(div_mod_int(A_64),B_52))),C_31) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_64),C_31) ).

tff(fact_2355_div__mod__equality,axiom,
    ! [A_64: nat,B_52: nat,C_31: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_64),B_52)),B_52)),hAPP_nat_nat(div_mod_nat(A_64),B_52))),C_31) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_64),C_31) ).

tff(fact_2356_div__mod__equality,axiom,
    ! [A_64: code_code_numeral,B_52: code_code_numeral,C_31: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_64),B_52)),B_52)),hAPP_C498520661umeral(div_mo1740067990umeral(A_64),B_52))),C_31) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_64),C_31) ).

tff(fact_2357_div__mod__equality,axiom,
    ! [A_64: quickcheck_code_int,B_52: quickcheck_code_int,C_31: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_64),B_52)),B_52)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_64),B_52))),C_31) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_64),C_31) ).

tff(fact_2358_zmod__zdiv__equality,axiom,
    ! [A_3: int,B_2: int] : A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2)) ).

tff(fact_2359_zdiv__zmult1__eq,axiom,
    ! [A_3: int,B_2: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) ).

tff(fact_2360_zdiv__zmod__equality,axiom,
    ! [B_2: int,A_3: int,K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) ).

tff(fact_2361_zdiv__zmod__equality2,axiom,
    ! [A_3: int,B_2: int,K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) ).

tff(fact_2362_zmod__zdiv__equality_H,axiom,
    ! [M: int,N: int] : hAPP_int_int(div_mod_int(M),N) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N)),N)) ).

tff(fact_2363_zmult__div__cancel,axiom,
    ! [N: int,M: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(div_mod_int(M),N)) ).

tff(fact_2364_mod__self,axiom,
    ! [A_63: int] : hAPP_int_int(div_mod_int(A_63),A_63) = zero_zero_int ).

tff(fact_2365_mod__self,axiom,
    ! [A_63: nat] : hAPP_nat_nat(div_mod_nat(A_63),A_63) = zero_zero_nat ).

tff(fact_2366_mod__self,axiom,
    ! [A_63: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(A_63),A_63) = zero_z126310315umeral ).

tff(fact_2367_mod__self,axiom,
    ! [A_63: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_63),A_63) = zero_z891286103de_int ).

tff(fact_2368_mod__by__0,axiom,
    ! [A_62: int] : hAPP_int_int(div_mod_int(A_62),zero_zero_int) = A_62 ).

tff(fact_2369_mod__by__0,axiom,
    ! [A_62: nat] : hAPP_nat_nat(div_mod_nat(A_62),zero_zero_nat) = A_62 ).

tff(fact_2370_mod__by__0,axiom,
    ! [A_62: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(A_62),zero_z126310315umeral) = A_62 ).

tff(fact_2371_mod__by__0,axiom,
    ! [A_62: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_62),zero_z891286103de_int) = A_62 ).

tff(fact_2372_mod__0,axiom,
    ! [A_61: int] : hAPP_int_int(div_mod_int(zero_zero_int),A_61) = zero_zero_int ).

tff(fact_2373_mod__0,axiom,
    ! [A_61: nat] : hAPP_nat_nat(div_mod_nat(zero_zero_nat),A_61) = zero_zero_nat ).

tff(fact_2374_mod__0,axiom,
    ! [A_61: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(zero_z126310315umeral),A_61) = zero_z126310315umeral ).

tff(fact_2375_mod__0,axiom,
    ! [A_61: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(zero_z891286103de_int),A_61) = zero_z891286103de_int ).

tff(fact_2376_negDivAlg__div__mod,axiom,
    ! [L_2: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L_2))
       => ( negDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) ) ) ) ).

tff(fact_2377_mod__mult__right__eq,axiom,
    ! [A_60: int,B_51: int,C_30: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_60),B_51)),C_30) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_60),hAPP_int_int(div_mod_int(B_51),C_30))),C_30) ).

tff(fact_2378_mod__mult__right__eq,axiom,
    ! [A_60: nat,B_51: nat,C_30: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_60),B_51)),C_30) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_60),hAPP_nat_nat(div_mod_nat(B_51),C_30))),C_30) ).

tff(fact_2379_mod__mult__right__eq,axiom,
    ! [A_60: code_code_numeral,B_51: code_code_numeral,C_30: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_60),B_51)),C_30) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_60),hAPP_C498520661umeral(div_mo1740067990umeral(B_51),C_30))),C_30) ).

tff(fact_2380_mod__mult__right__eq,axiom,
    ! [A_60: quickcheck_code_int,B_51: quickcheck_code_int,C_30: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_60),B_51)),C_30) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_60),hAPP_Q1762011733de_int(div_mo231679042de_int(B_51),C_30))),C_30) ).

tff(fact_2381_mod__mult__left__eq,axiom,
    ! [A_59: int,B_50: int,C_29: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_59),B_50)),C_29) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_59),C_29)),B_50)),C_29) ).

tff(fact_2382_mod__mult__left__eq,axiom,
    ! [A_59: nat,B_50: nat,C_29: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_59),B_50)),C_29) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_59),C_29)),B_50)),C_29) ).

tff(fact_2383_mod__mult__left__eq,axiom,
    ! [A_59: code_code_numeral,B_50: code_code_numeral,C_29: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_59),B_50)),C_29) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_59),C_29)),B_50)),C_29) ).

tff(fact_2384_mod__mult__left__eq,axiom,
    ! [A_59: quickcheck_code_int,B_50: quickcheck_code_int,C_29: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_59),B_50)),C_29) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_59),C_29)),B_50)),C_29) ).

tff(fact_2385_mod__mult__eq,axiom,
    ! [A_58: int,B_49: int,C_28: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_58),B_49)),C_28) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_58),C_28)),hAPP_int_int(div_mod_int(B_49),C_28))),C_28) ).

tff(fact_2386_mod__mult__eq,axiom,
    ! [A_58: nat,B_49: nat,C_28: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_58),B_49)),C_28) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_58),C_28)),hAPP_nat_nat(div_mod_nat(B_49),C_28))),C_28) ).

tff(fact_2387_mod__mult__eq,axiom,
    ! [A_58: code_code_numeral,B_49: code_code_numeral,C_28: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_58),B_49)),C_28) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_58),C_28)),hAPP_C498520661umeral(div_mo1740067990umeral(B_49),C_28))),C_28) ).

tff(fact_2388_mod__mult__eq,axiom,
    ! [A_58: quickcheck_code_int,B_49: quickcheck_code_int,C_28: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_58),B_49)),C_28) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_58),C_28)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_49),C_28))),C_28) ).

tff(fact_2389_mod__mult__mult1,axiom,
    ! [C_27: int,A_57: int,B_48: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_27),A_57)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_27),B_48)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_27),hAPP_int_int(div_mod_int(A_57),B_48)) ).

tff(fact_2390_mod__mult__mult1,axiom,
    ! [C_27: nat,A_57: nat,B_48: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_27),A_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_27),B_48)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_27),hAPP_nat_nat(div_mod_nat(A_57),B_48)) ).

tff(fact_2391_mod__mult__mult1,axiom,
    ! [C_27: code_code_numeral,A_57: code_code_numeral,B_48: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_27),A_57)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_27),B_48)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_27),hAPP_C498520661umeral(div_mo1740067990umeral(A_57),B_48)) ).

tff(fact_2392_mod__mult__mult1,axiom,
    ! [C_27: quickcheck_code_int,A_57: quickcheck_code_int,B_48: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_27),A_57)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_27),B_48)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_27),hAPP_Q1762011733de_int(div_mo231679042de_int(A_57),B_48)) ).

tff(fact_2393_mod__mult__mult2,axiom,
    ! [A_56: int,C_26: int,B_47: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_56),C_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_47),C_26)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_56),B_47)),C_26) ).

tff(fact_2394_mod__mult__mult2,axiom,
    ! [A_56: nat,C_26: nat,B_47: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_56),C_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_47),C_26)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_56),B_47)),C_26) ).

tff(fact_2395_mod__mult__mult2,axiom,
    ! [A_56: code_code_numeral,C_26: code_code_numeral,B_47: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_56),C_26)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_47),C_26)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_56),B_47)),C_26) ).

tff(fact_2396_mod__mult__mult2,axiom,
    ! [A_56: quickcheck_code_int,C_26: quickcheck_code_int,B_47: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_56),C_26)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_47),C_26)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_56),B_47)),C_26) ).

tff(fact_2397_zmod__simps_I4_J,axiom,
    ! [A_55: int,C_25: int,B_46: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_55),C_25)),B_46)),C_25) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_55),B_46)),C_25) ).

tff(fact_2398_zmod__simps_I4_J,axiom,
    ! [A_55: nat,C_25: nat,B_46: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_55),C_25)),B_46)),C_25) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_55),B_46)),C_25) ).

tff(fact_2399_zmod__simps_I4_J,axiom,
    ! [A_55: code_code_numeral,C_25: code_code_numeral,B_46: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_55),C_25)),B_46)),C_25) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_55),B_46)),C_25) ).

tff(fact_2400_zmod__simps_I4_J,axiom,
    ! [A_55: quickcheck_code_int,C_25: quickcheck_code_int,B_46: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_55),C_25)),B_46)),C_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_55),B_46)),C_25) ).

tff(fact_2401_mod__mult__cong,axiom,
    ! [B_45: int,B_44: int,A_54: int,C_24: int,A_53: int] :
      ( ( hAPP_int_int(div_mod_int(A_54),C_24) = hAPP_int_int(div_mod_int(A_53),C_24) )
     => ( ( hAPP_int_int(div_mod_int(B_45),C_24) = hAPP_int_int(div_mod_int(B_44),C_24) )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_54),B_45)),C_24) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_53),B_44)),C_24) ) ) ) ).

tff(fact_2402_mod__mult__cong,axiom,
    ! [B_45: nat,B_44: nat,A_54: nat,C_24: nat,A_53: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(A_54),C_24) = hAPP_nat_nat(div_mod_nat(A_53),C_24) )
     => ( ( hAPP_nat_nat(div_mod_nat(B_45),C_24) = hAPP_nat_nat(div_mod_nat(B_44),C_24) )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_54),B_45)),C_24) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_53),B_44)),C_24) ) ) ) ).

tff(fact_2403_mod__mult__cong,axiom,
    ! [B_45: code_code_numeral,B_44: code_code_numeral,A_54: code_code_numeral,C_24: code_code_numeral,A_53: code_code_numeral] :
      ( ( hAPP_C498520661umeral(div_mo1740067990umeral(A_54),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(A_53),C_24) )
     => ( ( hAPP_C498520661umeral(div_mo1740067990umeral(B_45),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(B_44),C_24) )
       => ( hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_54),B_45)),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_53),B_44)),C_24) ) ) ) ).

tff(fact_2404_mod__mult__cong,axiom,
    ! [B_45: quickcheck_code_int,B_44: quickcheck_code_int,A_54: quickcheck_code_int,C_24: quickcheck_code_int,A_53: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(A_54),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_53),C_24) )
     => ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_45),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_44),C_24) )
       => ( hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_54),B_45)),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_53),B_44)),C_24) ) ) ) ).

tff(fact_2405_mod__add__cong,axiom,
    ! [B_43: int,B_42: int,A_52: int,C_23: int,A_51: int] :
      ( ( hAPP_int_int(div_mod_int(A_52),C_23) = hAPP_int_int(div_mod_int(A_51),C_23) )
     => ( ( hAPP_int_int(div_mod_int(B_43),C_23) = hAPP_int_int(div_mod_int(B_42),C_23) )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_52),B_43)),C_23) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_51),B_42)),C_23) ) ) ) ).

tff(fact_2406_mod__add__cong,axiom,
    ! [B_43: nat,B_42: nat,A_52: nat,C_23: nat,A_51: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(A_52),C_23) = hAPP_nat_nat(div_mod_nat(A_51),C_23) )
     => ( ( hAPP_nat_nat(div_mod_nat(B_43),C_23) = hAPP_nat_nat(div_mod_nat(B_42),C_23) )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_52),B_43)),C_23) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_51),B_42)),C_23) ) ) ) ).

tff(fact_2407_mod__add__cong,axiom,
    ! [B_43: code_code_numeral,B_42: code_code_numeral,A_52: code_code_numeral,C_23: code_code_numeral,A_51: code_code_numeral] :
      ( ( hAPP_C498520661umeral(div_mo1740067990umeral(A_52),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(A_51),C_23) )
     => ( ( hAPP_C498520661umeral(div_mo1740067990umeral(B_43),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(B_42),C_23) )
       => ( hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_52),B_43)),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_51),B_42)),C_23) ) ) ) ).

tff(fact_2408_mod__add__cong,axiom,
    ! [B_43: quickcheck_code_int,B_42: quickcheck_code_int,A_52: quickcheck_code_int,C_23: quickcheck_code_int,A_51: quickcheck_code_int] :
      ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(A_52),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_51),C_23) )
     => ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_43),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_42),C_23) )
       => ( hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_52),B_43)),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_51),B_42)),C_23) ) ) ) ).

tff(fact_2409_zmod__simps_I1_J,axiom,
    ! [A_50: int,C_22: int,B_41: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_50),C_22)),B_41)),C_22) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_50),B_41)),C_22) ).

tff(fact_2410_zmod__simps_I1_J,axiom,
    ! [A_50: nat,C_22: nat,B_41: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_50),C_22)),B_41)),C_22) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_50),B_41)),C_22) ).

tff(fact_2411_zmod__simps_I1_J,axiom,
    ! [A_50: code_code_numeral,C_22: code_code_numeral,B_41: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_50),C_22)),B_41)),C_22) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_50),B_41)),C_22) ).

tff(fact_2412_zmod__simps_I1_J,axiom,
    ! [A_50: quickcheck_code_int,C_22: quickcheck_code_int,B_41: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),C_22)),B_41)),C_22) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_50),B_41)),C_22) ).

tff(fact_2413_zmod__simps_I2_J,axiom,
    ! [A_49: int,B_40: int,C_21: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_49),hAPP_int_int(div_mod_int(B_40),C_21))),C_21) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_49),B_40)),C_21) ).

tff(fact_2414_zmod__simps_I2_J,axiom,
    ! [A_49: nat,B_40: nat,C_21: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_49),hAPP_nat_nat(div_mod_nat(B_40),C_21))),C_21) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_49),B_40)),C_21) ).

tff(fact_2415_zmod__simps_I2_J,axiom,
    ! [A_49: code_code_numeral,B_40: code_code_numeral,C_21: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_49),hAPP_C498520661umeral(div_mo1740067990umeral(B_40),C_21))),C_21) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_49),B_40)),C_21) ).

tff(fact_2416_zmod__simps_I2_J,axiom,
    ! [A_49: quickcheck_code_int,B_40: quickcheck_code_int,C_21: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_49),hAPP_Q1762011733de_int(div_mo231679042de_int(B_40),C_21))),C_21) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_49),B_40)),C_21) ).

tff(fact_2417_mod__add__eq,axiom,
    ! [A_48: int,B_39: int,C_20: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_48),B_39)),C_20) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_48),C_20)),hAPP_int_int(div_mod_int(B_39),C_20))),C_20) ).

tff(fact_2418_mod__add__eq,axiom,
    ! [A_48: nat,B_39: nat,C_20: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_48),B_39)),C_20) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_48),C_20)),hAPP_nat_nat(div_mod_nat(B_39),C_20))),C_20) ).

tff(fact_2419_mod__add__eq,axiom,
    ! [A_48: code_code_numeral,B_39: code_code_numeral,C_20: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_48),B_39)),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_48),C_20)),hAPP_C498520661umeral(div_mo1740067990umeral(B_39),C_20))),C_20) ).

tff(fact_2420_mod__add__eq,axiom,
    ! [A_48: quickcheck_code_int,B_39: quickcheck_code_int,C_20: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_48),B_39)),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_48),C_20)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_39),C_20))),C_20) ).

tff(fact_2421_mod__add__left__eq,axiom,
    ! [A_47: int,B_38: int,C_19: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_47),B_38)),C_19) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_47),C_19)),B_38)),C_19) ).

tff(fact_2422_mod__add__left__eq,axiom,
    ! [A_47: nat,B_38: nat,C_19: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_47),B_38)),C_19) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_47),C_19)),B_38)),C_19) ).

tff(fact_2423_mod__add__left__eq,axiom,
    ! [A_47: code_code_numeral,B_38: code_code_numeral,C_19: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_47),B_38)),C_19) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_47),C_19)),B_38)),C_19) ).

tff(fact_2424_mod__add__left__eq,axiom,
    ! [A_47: quickcheck_code_int,B_38: quickcheck_code_int,C_19: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_47),B_38)),C_19) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_47),C_19)),B_38)),C_19) ).

tff(fact_2425_mod__add__right__eq,axiom,
    ! [A_46: int,B_37: int,C_18: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),B_37)),C_18) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),hAPP_int_int(div_mod_int(B_37),C_18))),C_18) ).

tff(fact_2426_mod__add__right__eq,axiom,
    ! [A_46: nat,B_37: nat,C_18: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_46),B_37)),C_18) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_46),hAPP_nat_nat(div_mod_nat(B_37),C_18))),C_18) ).

tff(fact_2427_mod__add__right__eq,axiom,
    ! [A_46: code_code_numeral,B_37: code_code_numeral,C_18: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_46),B_37)),C_18) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_46),hAPP_C498520661umeral(div_mo1740067990umeral(B_37),C_18))),C_18) ).

tff(fact_2428_mod__add__right__eq,axiom,
    ! [A_46: quickcheck_code_int,B_37: quickcheck_code_int,C_18: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_46),B_37)),C_18) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_46),hAPP_Q1762011733de_int(div_mo231679042de_int(B_37),C_18))),C_18) ).

tff(fact_2429_mod__add__self1,axiom,
    ! [B_36: int,A_45: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_36),A_45)),B_36) = hAPP_int_int(div_mod_int(A_45),B_36) ).

tff(fact_2430_mod__add__self1,axiom,
    ! [B_36: nat,A_45: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_36),A_45)),B_36) = hAPP_nat_nat(div_mod_nat(A_45),B_36) ).

tff(fact_2431_mod__add__self1,axiom,
    ! [B_36: code_code_numeral,A_45: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_36),A_45)),B_36) = hAPP_C498520661umeral(div_mo1740067990umeral(A_45),B_36) ).

tff(fact_2432_mod__add__self1,axiom,
    ! [B_36: quickcheck_code_int,A_45: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_36),A_45)),B_36) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_45),B_36) ).

tff(fact_2433_mod__add__self2,axiom,
    ! [A_44: int,B_35: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_44),B_35)),B_35) = hAPP_int_int(div_mod_int(A_44),B_35) ).

tff(fact_2434_mod__add__self2,axiom,
    ! [A_44: nat,B_35: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_44),B_35)),B_35) = hAPP_nat_nat(div_mod_nat(A_44),B_35) ).

tff(fact_2435_mod__add__self2,axiom,
    ! [A_44: code_code_numeral,B_35: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_44),B_35)),B_35) = hAPP_C498520661umeral(div_mo1740067990umeral(A_44),B_35) ).

tff(fact_2436_mod__add__self2,axiom,
    ! [A_44: quickcheck_code_int,B_35: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_44),B_35)),B_35) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_44),B_35) ).

tff(fact_2437_mod__diff__cong,axiom,
    ! [B_34: int,B_33: int,A_43: int,C_17: int,A_42: int] :
      ( ( hAPP_int_int(div_mod_int(A_43),C_17) = hAPP_int_int(div_mod_int(A_42),C_17) )
     => ( ( hAPP_int_int(div_mod_int(B_34),C_17) = hAPP_int_int(div_mod_int(B_33),C_17) )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_43),B_34)),C_17) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_42),B_33)),C_17) ) ) ) ).

tff(fact_2438_mod__diff__eq,axiom,
    ! [A_41: int,B_32: int,C_16: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_41),B_32)),C_16) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_41),C_16)),hAPP_int_int(div_mod_int(B_32),C_16))),C_16) ).

tff(fact_2439_mod__diff__left__eq,axiom,
    ! [A_40: int,B_31: int,C_15: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_40),B_31)),C_15) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_40),C_15)),B_31)),C_15) ).

tff(fact_2440_mod__diff__right__eq,axiom,
    ! [A_39: int,B_30: int,C_14: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_39),B_30)),C_14) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_39),hAPP_int_int(div_mod_int(B_30),C_14))),C_14) ).

tff(fact_2441_dvd__mod__iff,axiom,
    ! [Ma: code_code_numeral,K_1: code_code_numeral,N_1: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),N_1))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),hAPP_C498520661umeral(div_mo1740067990umeral(Ma),N_1)))
      <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),Ma)) ) ) ).

tff(fact_2442_dvd__mod__iff,axiom,
    ! [Ma: quickcheck_code_int,K_1: quickcheck_code_int,N_1: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),N_1))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),hAPP_Q1762011733de_int(div_mo231679042de_int(Ma),N_1)))
      <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),Ma)) ) ) ).

tff(fact_2443_dvd__mod__iff,axiom,
    ! [Ma: int,K_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(div_mod_int(Ma),N_1)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) ) ) ).

tff(fact_2444_dvd__mod__iff,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(div_mod_nat(Ma),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) ) ) ).

tff(fact_2445_mod__mod__cancel,axiom,
    ! [A_38: code_code_numeral,C_13: code_code_numeral,B_29: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_13),B_29))
     => ( hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_38),B_29)),C_13) = hAPP_C498520661umeral(div_mo1740067990umeral(A_38),C_13) ) ) ).

tff(fact_2446_mod__mod__cancel,axiom,
    ! [A_38: quickcheck_code_int,C_13: quickcheck_code_int,B_29: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_13),B_29))
     => ( hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_38),B_29)),C_13) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_38),C_13) ) ) ).

tff(fact_2447_mod__mod__cancel,axiom,
    ! [A_38: int,C_13: int,B_29: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_13),B_29))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_38),B_29)),C_13) = hAPP_int_int(div_mod_int(A_38),C_13) ) ) ).

tff(fact_2448_mod__mod__cancel,axiom,
    ! [A_38: nat,C_13: nat,B_29: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_13),B_29))
     => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_38),B_29)),C_13) = hAPP_nat_nat(div_mod_nat(A_38),C_13) ) ) ).

tff(fact_2449_dvd__mod,axiom,
    ! [N_29: code_code_numeral,K_6: code_code_numeral,M_16: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),M_16))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),N_29))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),hAPP_C498520661umeral(div_mo1740067990umeral(M_16),N_29))) ) ) ).

tff(fact_2450_dvd__mod,axiom,
    ! [N_29: quickcheck_code_int,K_6: quickcheck_code_int,M_16: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),M_16))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),N_29))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),hAPP_Q1762011733de_int(div_mo231679042de_int(M_16),N_29))) ) ) ).

tff(fact_2451_dvd__mod,axiom,
    ! [N_29: int,K_6: int,M_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),M_16))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),N_29))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),hAPP_int_int(div_mod_int(M_16),N_29))) ) ) ).

tff(fact_2452_dvd__mod,axiom,
    ! [N_29: nat,K_6: nat,M_16: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),M_16))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),N_29))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),hAPP_nat_nat(div_mod_nat(M_16),N_29))) ) ) ).

tff(fact_2453_dvd__mod__imp__dvd,axiom,
    ! [K_5: code_code_numeral,M_15: code_code_numeral,N_28: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),hAPP_C498520661umeral(div_mo1740067990umeral(M_15),N_28)))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),N_28))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),M_15)) ) ) ).

tff(fact_2454_dvd__mod__imp__dvd,axiom,
    ! [K_5: quickcheck_code_int,M_15: quickcheck_code_int,N_28: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),hAPP_Q1762011733de_int(div_mo231679042de_int(M_15),N_28)))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),N_28))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),M_15)) ) ) ).

tff(fact_2455_dvd__mod__imp__dvd,axiom,
    ! [K_5: int,M_15: int,N_28: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),hAPP_int_int(div_mod_int(M_15),N_28)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),N_28))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),M_15)) ) ) ).

tff(fact_2456_dvd__mod__imp__dvd,axiom,
    ! [K_5: nat,M_15: nat,N_28: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),hAPP_nat_nat(div_mod_nat(M_15),N_28)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),N_28))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),M_15)) ) ) ).

tff(fact_2457_zdiv__zero,axiom,
    ! [B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),B_2) = zero_zero_int ).

tff(fact_2458_zmod__self,axiom,
    ! [A_3: int] : hAPP_int_int(div_mod_int(A_3),A_3) = zero_zero_int ).

tff(fact_2459_zmod__zero,axiom,
    ! [B_2: int] : hAPP_int_int(div_mod_int(zero_zero_int),B_2) = zero_zero_int ).

tff(fact_2460_zmod__zmult1__eq,axiom,
    ! [A_3: int,B_2: int,C_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) ).

tff(fact_2461_zmod__simps_I3_J,axiom,
    ! [A_3: int,B_2: int,C_1: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) ).

tff(fact_2462_zdiff__zmod__left,axiom,
    ! [X: int,M: int,Y: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(X),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M) ).

tff(fact_2463_zdiff__zmod__right,axiom,
    ! [X: int,Y: int,M: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(div_mod_int(Y),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M) ).

tff(fact_2464_mod__mod__is__mod,axiom,
    ! [X: int,M: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(div_mod_int(X),M)),M)) ).

tff(fact_2465_Residues_Oaux,axiom,
    ! [X: int,M: int,Y: int] :
      ( ( hAPP_int_int(div_mod_int(X),M) = hAPP_int_int(div_mod_int(Y),M) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)) ) ).

tff(fact_2466_zcong__zmod,axiom,
    ! [A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(div_mod_int(A),Ma)),hAPP_int_int(div_mod_int(B),Ma)),Ma)) ) ).

tff(fact_2467_zdvd__zmod,axiom,
    ! [N: int,F_3: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M),N))) ) ) ).

tff(fact_2468_zdvd__zmod__imp__zdvd,axiom,
    ! [K_2: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(div_mod_int(M),N)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).

tff(fact_2469_zpower__zmod,axiom,
    ! [X: int,M: int,Y: nat] : hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(div_mod_int(X),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),M) ).

tff(fact_2470_div__by__0,axiom,
    ! [A_37: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_37),zero_zero_int) = zero_zero_int ).

tff(fact_2471_div__by__0,axiom,
    ! [A_37: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_37),zero_zero_nat) = zero_zero_nat ).

tff(fact_2472_div__by__0,axiom,
    ! [A_37: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_37),zero_z126310315umeral) = zero_z126310315umeral ).

tff(fact_2473_div__by__0,axiom,
    ! [A_37: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_37),zero_z891286103de_int) = zero_z891286103de_int ).

tff(fact_2474_div__0,axiom,
    ! [A_36: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),A_36) = zero_zero_int ).

tff(fact_2475_div__0,axiom,
    ! [A_36: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,zero_zero_nat),A_36) = zero_zero_nat ).

tff(fact_2476_div__0,axiom,
    ! [A_36: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,zero_z126310315umeral),A_36) = zero_z126310315umeral ).

tff(fact_2477_div__0,axiom,
    ! [A_36: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,zero_z891286103de_int),A_36) = zero_z891286103de_int ).

tff(fact_2478_div__by__1,axiom,
    ! [A_35: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_35),one_one_int) = A_35 ).

tff(fact_2479_div__by__1,axiom,
    ! [A_35: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_35),one_one_nat) = A_35 ).

tff(fact_2480_div__by__1,axiom,
    ! [A_35: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_35),one_on1645066479umeral) = A_35 ).

tff(fact_2481_div__by__1,axiom,
    ! [A_35: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_35),one_on1684967323de_int) = A_35 ).

tff(fact_2482_div__dvd__div,axiom,
    ! [C: code_code_numeral,A: code_code_numeral,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),C))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,C),A)))
        <=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B),C)) ) ) ) ).

tff(fact_2483_div__dvd__div,axiom,
    ! [C: quickcheck_code_int,A: quickcheck_code_int,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),C))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,C),A)))
        <=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B),C)) ) ) ) ).

tff(fact_2484_div__dvd__div,axiom,
    ! [C: int,A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),C))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,C),A)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B),C)) ) ) ) ).

tff(fact_2485_div__dvd__div,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,C),A)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C)) ) ) ) ).

tff(fact_2486_xzgcda__linear__aux2,axiom,
    ! [R_3: int,S_1: int,T_1: int,R_5: int,S_3: int,M: int,T_2: int,N: int] :
      ( ( R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N)) )
     => ( ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)) )
       => ( hAPP_int_int(div_mod_int(R_5),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1))),N)) ) ) ) ).

tff(fact_2487_zcong__zmod__aux,axiom,
    ! [A_3: int,B_2: int,M: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),M)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),M)),hAPP_int_int(div_mod_int(B_2),M))) ).

tff(fact_2488_EvenOdd_Oodd__times__odd,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zOdd)) ) ) ).

tff(fact_2489_odd__mult__odd__prop,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zOdd))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) ) ).

tff(fact_2490_StandardRes__prop1,axiom,
    ! [X: int,M: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(standardRes(M),X)),M)) ).

tff(fact_2491_mod__mult__self1__is__0,axiom,
    ! [B_28: int,A_34: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_28),A_34)),B_28) = zero_zero_int ).

tff(fact_2492_mod__mult__self1__is__0,axiom,
    ! [B_28: nat,A_34: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_28),A_34)),B_28) = zero_zero_nat ).

tff(fact_2493_mod__mult__self1__is__0,axiom,
    ! [B_28: code_code_numeral,A_34: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_28),A_34)),B_28) = zero_z126310315umeral ).

tff(fact_2494_mod__mult__self1__is__0,axiom,
    ! [B_28: quickcheck_code_int,A_34: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_28),A_34)),B_28) = zero_z891286103de_int ).

tff(fact_2495_mod__mult__self2__is__0,axiom,
    ! [A_33: int,B_27: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_33),B_27)),B_27) = zero_zero_int ).

tff(fact_2496_mod__mult__self2__is__0,axiom,
    ! [A_33: nat,B_27: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_33),B_27)),B_27) = zero_zero_nat ).

tff(fact_2497_mod__mult__self2__is__0,axiom,
    ! [A_33: code_code_numeral,B_27: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_33),B_27)),B_27) = zero_z126310315umeral ).

tff(fact_2498_mod__mult__self2__is__0,axiom,
    ! [A_33: quickcheck_code_int,B_27: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_33),B_27)),B_27) = zero_z891286103de_int ).

tff(fact_2499_mod__by__1,axiom,
    ! [A_32: int] : hAPP_int_int(div_mod_int(A_32),one_one_int) = zero_zero_int ).

tff(fact_2500_mod__by__1,axiom,
    ! [A_32: nat] : hAPP_nat_nat(div_mod_nat(A_32),one_one_nat) = zero_zero_nat ).

tff(fact_2501_mod__by__1,axiom,
    ! [A_32: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(A_32),one_on1645066479umeral) = zero_z126310315umeral ).

tff(fact_2502_mod__by__1,axiom,
    ! [A_32: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(A_32),one_on1684967323de_int) = zero_z891286103de_int ).

tff(fact_2503_zmod__zmult2__eq,axiom,
    ! [A_3: int,B_2: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hAPP_int_int(div_mod_int(A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))),hAPP_int_int(div_mod_int(A_3),B_2)) ) ) ).

tff(fact_2504_mod__mult__self1,axiom,
    ! [A_31: int,C_12: int,B_26: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),B_26))),B_26) = hAPP_int_int(div_mod_int(A_31),B_26) ).

tff(fact_2505_mod__mult__self1,axiom,
    ! [A_31: nat,C_12: nat,B_26: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),B_26))),B_26) = hAPP_nat_nat(div_mod_nat(A_31),B_26) ).

tff(fact_2506_mod__mult__self1,axiom,
    ! [A_31: code_code_numeral,C_12: code_code_numeral,B_26: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_31),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),B_26))),B_26) = hAPP_C498520661umeral(div_mo1740067990umeral(A_31),B_26) ).

tff(fact_2507_mod__mult__self1,axiom,
    ! [A_31: quickcheck_code_int,C_12: quickcheck_code_int,B_26: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_31),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),B_26))),B_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_31),B_26) ).

tff(fact_2508_mod__mult__self2,axiom,
    ! [A_30: int,B_25: int,C_11: int] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_25),C_11))),B_25) = hAPP_int_int(div_mod_int(A_30),B_25) ).

tff(fact_2509_mod__mult__self2,axiom,
    ! [A_30: nat,B_25: nat,C_11: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_25),C_11))),B_25) = hAPP_nat_nat(div_mod_nat(A_30),B_25) ).

tff(fact_2510_mod__mult__self2,axiom,
    ! [A_30: code_code_numeral,B_25: code_code_numeral,C_11: code_code_numeral] : hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_30),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_25),C_11))),B_25) = hAPP_C498520661umeral(div_mo1740067990umeral(A_30),B_25) ).

tff(fact_2511_mod__mult__self2,axiom,
    ! [A_30: quickcheck_code_int,B_25: quickcheck_code_int,C_11: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_30),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_25),C_11))),B_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_30),B_25) ).

tff(fact_2512_dvd__imp__mod__0,axiom,
    ! [A_29: code_code_numeral,B_24: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_29),B_24))
     => ( hAPP_C498520661umeral(div_mo1740067990umeral(B_24),A_29) = zero_z126310315umeral ) ) ).

tff(fact_2513_dvd__imp__mod__0,axiom,
    ! [A_29: quickcheck_code_int,B_24: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_29),B_24))
     => ( hAPP_Q1762011733de_int(div_mo231679042de_int(B_24),A_29) = zero_z891286103de_int ) ) ).

tff(fact_2514_dvd__imp__mod__0,axiom,
    ! [A_29: int,B_24: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_29),B_24))
     => ( hAPP_int_int(div_mod_int(B_24),A_29) = zero_zero_int ) ) ).

tff(fact_2515_dvd__imp__mod__0,axiom,
    ! [A_29: nat,B_24: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_29),B_24))
     => ( hAPP_nat_nat(div_mod_nat(B_24),A_29) = zero_zero_nat ) ) ).

tff(fact_2516_dvd__eq__mod__eq__0,axiom,
    ! [A: code_code_numeral,B: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
    <=> ( hAPP_C498520661umeral(div_mo1740067990umeral(B),A) = zero_z126310315umeral ) ) ).

tff(fact_2517_dvd__eq__mod__eq__0,axiom,
    ! [A: quickcheck_code_int,B: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
    <=> ( hAPP_Q1762011733de_int(div_mo231679042de_int(B),A) = zero_z891286103de_int ) ) ).

tff(fact_2518_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
    <=> ( hAPP_int_int(div_mod_int(B),A) = zero_zero_int ) ) ).

tff(fact_2519_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
    <=> ( hAPP_nat_nat(div_mod_nat(B),A) = zero_zero_nat ) ) ).

tff(fact_2520_div__neg__pos__less0,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int)) ) ) ).

tff(fact_2521_neg__imp__zdiv__neg__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) ) ) ).

tff(fact_2522_pos__imp__zdiv__neg__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ) ).

tff(fact_2523_zdiv__self,axiom,
    ! [A_3: int] :
      ( ( A_3 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),A_3) = one_one_int ) ) ).

tff(fact_2524_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))) ) ) ).

tff(fact_2525_zdiv__number__of__Bit0,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) ).

tff(fact_2526_neg__mod__bound,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))) ) ).

tff(fact_2527_pos__mod__bound,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) ) ).

tff(fact_2528_zmod__le__nonneg__dividend,axiom,
    ! [K_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M),K_2)),M)) ) ).

tff(fact_2529_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X),Y))) ) ) ).

tff(fact_2530_zmod__eq__0__iff,axiom,
    ! [Ma: int,D: int] :
      ( ( hAPP_int_int(div_mod_int(Ma),D) = zero_zero_int )
    <=> ? [Q_4: int] : Ma = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Q_4) ) ).

tff(fact_2531_zdvd__mult__div__cancel,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N)) = M ) ) ).

tff(fact_2532_zmod__eq__dvd__iff,axiom,
    ! [X_1: int,N_1: int,Y_2: int] :
      ( ( hAPP_int_int(div_mod_int(X_1),N_1) = hAPP_int_int(div_mod_int(Y_2),N_1) )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2))) ) ).

tff(fact_2533_div__mult__mult1,axiom,
    ! [A_28: int,B_23: int,C_10: int] :
      ( ( C_10 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),A_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),B_23)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_28),B_23) ) ) ).

tff(fact_2534_div__mult__mult1,axiom,
    ! [A_28: nat,B_23: nat,C_10: nat] :
      ( ( C_10 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),A_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),B_23)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_28),B_23) ) ) ).

tff(fact_2535_div__mult__mult1,axiom,
    ! [A_28: code_code_numeral,B_23: code_code_numeral,C_10: code_code_numeral] :
      ( ( C_10 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),A_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),B_23)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_28),B_23) ) ) ).

tff(fact_2536_div__mult__mult1,axiom,
    ! [A_28: quickcheck_code_int,B_23: quickcheck_code_int,C_10: quickcheck_code_int] :
      ( ( C_10 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),A_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),B_23)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_28),B_23) ) ) ).

tff(fact_2537_div__mult__mult2,axiom,
    ! [A_27: int,B_22: int,C_9: int] :
      ( ( C_9 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),C_9)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_22),C_9)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_27),B_22) ) ) ).

tff(fact_2538_div__mult__mult2,axiom,
    ! [A_27: nat,B_22: nat,C_9: nat] :
      ( ( C_9 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),C_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_22),C_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_27),B_22) ) ) ).

tff(fact_2539_div__mult__mult2,axiom,
    ! [A_27: code_code_numeral,B_22: code_code_numeral,C_9: code_code_numeral] :
      ( ( C_9 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_27),C_9)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_22),C_9)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_27),B_22) ) ) ).

tff(fact_2540_div__mult__mult2,axiom,
    ! [A_27: quickcheck_code_int,B_22: quickcheck_code_int,C_9: quickcheck_code_int] :
      ( ( C_9 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_27),C_9)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_22),C_9)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_27),B_22) ) ) ).

tff(fact_2541_div__mult__self1__is__id,axiom,
    ! [A_26: int,B_21: int] :
      ( ( B_21 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_21),A_26)),B_21) = A_26 ) ) ).

tff(fact_2542_div__mult__self1__is__id,axiom,
    ! [A_26: nat,B_21: nat] :
      ( ( B_21 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_21),A_26)),B_21) = A_26 ) ) ).

tff(fact_2543_div__mult__self1__is__id,axiom,
    ! [A_26: code_code_numeral,B_21: code_code_numeral] :
      ( ( B_21 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_21),A_26)),B_21) = A_26 ) ) ).

tff(fact_2544_div__mult__self1__is__id,axiom,
    ! [A_26: quickcheck_code_int,B_21: quickcheck_code_int] :
      ( ( B_21 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_21),A_26)),B_21) = A_26 ) ) ).

tff(fact_2545_div__mult__self2__is__id,axiom,
    ! [A_25: int,B_20: int] :
      ( ( B_20 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_25),B_20)),B_20) = A_25 ) ) ).

tff(fact_2546_div__mult__self2__is__id,axiom,
    ! [A_25: nat,B_20: nat] :
      ( ( B_20 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_25),B_20)),B_20) = A_25 ) ) ).

tff(fact_2547_div__mult__self2__is__id,axiom,
    ! [A_25: code_code_numeral,B_20: code_code_numeral] :
      ( ( B_20 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_25),B_20)),B_20) = A_25 ) ) ).

tff(fact_2548_div__mult__self2__is__id,axiom,
    ! [A_25: quickcheck_code_int,B_20: quickcheck_code_int] :
      ( ( B_20 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_25),B_20)),B_20) = A_25 ) ) ).

tff(fact_2549_div__mult__mult1__if,axiom,
    ! [A_24: int,B_19: int,C_8: int] :
      ( ( ( C_8 = zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),A_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),B_19)) = zero_zero_int ) )
      & ( ( C_8 != zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),A_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),B_19)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_24),B_19) ) ) ) ).

tff(fact_2550_div__mult__mult1__if,axiom,
    ! [A_24: nat,B_19: nat,C_8: nat] :
      ( ( ( C_8 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),A_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),B_19)) = zero_zero_nat ) )
      & ( ( C_8 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),A_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),B_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_24),B_19) ) ) ) ).

tff(fact_2551_div__mult__mult1__if,axiom,
    ! [A_24: code_code_numeral,B_19: code_code_numeral,C_8: code_code_numeral] :
      ( ( ( C_8 = zero_z126310315umeral )
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),A_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),B_19)) = zero_z126310315umeral ) )
      & ( ( C_8 != zero_z126310315umeral )
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),A_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),B_19)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_24),B_19) ) ) ) ).

tff(fact_2552_div__mult__mult1__if,axiom,
    ! [A_24: quickcheck_code_int,B_19: quickcheck_code_int,C_8: quickcheck_code_int] :
      ( ( ( C_8 = zero_z891286103de_int )
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),A_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),B_19)) = zero_z891286103de_int ) )
      & ( ( C_8 != zero_z891286103de_int )
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),A_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),B_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_24),B_19) ) ) ) ).

tff(fact_2553_div__self,axiom,
    ! [A_23: int] :
      ( ( A_23 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_23),A_23) = one_one_int ) ) ).

tff(fact_2554_div__self,axiom,
    ! [A_23: nat] :
      ( ( A_23 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_23),A_23) = one_one_nat ) ) ).

tff(fact_2555_div__self,axiom,
    ! [A_23: code_code_numeral] :
      ( ( A_23 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_23),A_23) = one_on1645066479umeral ) ) ).

tff(fact_2556_div__self,axiom,
    ! [A_23: quickcheck_code_int] :
      ( ( A_23 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_23),A_23) = one_on1684967323de_int ) ) ).

tff(fact_2557_div__mult__div__if__dvd,axiom,
    ! [Z_7: code_code_numeral,W_2: code_code_numeral,Y_9: code_code_numeral,X_12: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_9),X_12))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_7),W_2))
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_12),Y_9)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,W_2),Z_7)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_12),W_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Y_9),Z_7)) ) ) ) ).

tff(fact_2558_div__mult__div__if__dvd,axiom,
    ! [Z_7: quickcheck_code_int,W_2: quickcheck_code_int,Y_9: quickcheck_code_int,X_12: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_9),X_12))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_7),W_2))
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_12),Y_9)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,W_2),Z_7)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_12),W_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Y_9),Z_7)) ) ) ) ).

tff(fact_2559_div__mult__div__if__dvd,axiom,
    ! [Z_7: int,W_2: int,Y_9: int,X_12: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_9),X_12))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_7),W_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_12),Y_9)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,W_2),Z_7)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_12),W_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_9),Z_7)) ) ) ) ).

tff(fact_2560_div__mult__div__if__dvd,axiom,
    ! [Z_7: nat,W_2: nat,Y_9: nat,X_12: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_9),X_12))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_7),W_2))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_12),Y_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,W_2),Z_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_12),W_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_9),Z_7)) ) ) ) ).

tff(fact_2561_dvd__div__mult,axiom,
    ! [C_7: code_code_numeral,A_22: code_code_numeral,B_18: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_22),B_18))
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_18),A_22)),C_7) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_18),C_7)),A_22) ) ) ).

tff(fact_2562_dvd__div__mult,axiom,
    ! [C_7: quickcheck_code_int,A_22: quickcheck_code_int,B_18: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_22),B_18))
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_18),A_22)),C_7) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_18),C_7)),A_22) ) ) ).

tff(fact_2563_dvd__div__mult,axiom,
    ! [C_7: int,A_22: int,B_18: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_22),B_18))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_18),A_22)),C_7) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_18),C_7)),A_22) ) ) ).

tff(fact_2564_dvd__div__mult,axiom,
    ! [C_7: nat,A_22: nat,B_18: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_22),B_18))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_18),A_22)),C_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_18),C_7)),A_22) ) ) ).

tff(fact_2565_dvd__div__mult__self,axiom,
    ! [A_21: code_code_numeral,B_17: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_21),B_17))
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_17),A_21)),A_21) = B_17 ) ) ).

tff(fact_2566_dvd__div__mult__self,axiom,
    ! [A_21: quickcheck_code_int,B_17: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_21),B_17))
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_17),A_21)),A_21) = B_17 ) ) ).

tff(fact_2567_dvd__div__mult__self,axiom,
    ! [A_21: int,B_17: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_21),B_17))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_17),A_21)),A_21) = B_17 ) ) ).

tff(fact_2568_dvd__div__mult__self,axiom,
    ! [A_21: nat,B_17: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_21),B_17))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_17),A_21)),A_21) = B_17 ) ) ).

tff(fact_2569_div__mult__swap,axiom,
    ! [A_20: code_code_numeral,C_6: code_code_numeral,B_16: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_6),B_16))
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_20),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_16),C_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_20),B_16)),C_6) ) ) ).

tff(fact_2570_div__mult__swap,axiom,
    ! [A_20: quickcheck_code_int,C_6: quickcheck_code_int,B_16: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_6),B_16))
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_20),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_16),C_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_20),B_16)),C_6) ) ) ).

tff(fact_2571_div__mult__swap,axiom,
    ! [A_20: int,C_6: int,B_16: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_6),B_16))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_20),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_16),C_6)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_20),B_16)),C_6) ) ) ).

tff(fact_2572_div__mult__swap,axiom,
    ! [A_20: nat,C_6: nat,B_16: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_6),B_16))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_20),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_16),C_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_20),B_16)),C_6) ) ) ).

tff(fact_2573_dvd__mult__div__cancel,axiom,
    ! [A_19: code_code_numeral,B_15: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_19),B_15))
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_19),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_15),A_19)) = B_15 ) ) ).

tff(fact_2574_dvd__mult__div__cancel,axiom,
    ! [A_19: quickcheck_code_int,B_15: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_19),B_15))
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_19),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_15),A_19)) = B_15 ) ) ).

tff(fact_2575_dvd__mult__div__cancel,axiom,
    ! [A_19: int,B_15: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_19),B_15))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_19),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_15),A_19)) = B_15 ) ) ).

tff(fact_2576_dvd__mult__div__cancel,axiom,
    ! [A_19: nat,B_15: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_19),B_15))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_19),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_15),A_19)) = B_15 ) ) ).

tff(fact_2577_div__add,axiom,
    ! [Y_8: code_code_numeral,Z_6: code_code_numeral,X_11: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),X_11))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),Y_8))
       => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_11),Y_8)),Z_6) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_11),Z_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,Y_8),Z_6)) ) ) ) ).

tff(fact_2578_div__add,axiom,
    ! [Y_8: quickcheck_code_int,Z_6: quickcheck_code_int,X_11: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),X_11))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),Y_8))
       => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_11),Y_8)),Z_6) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_11),Z_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,Y_8),Z_6)) ) ) ) ).

tff(fact_2579_div__add,axiom,
    ! [Y_8: int,Z_6: int,X_11: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),X_11))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),Y_8))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_11),Y_8)),Z_6) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_11),Z_6)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_8),Z_6)) ) ) ) ).

tff(fact_2580_div__add,axiom,
    ! [Y_8: nat,Z_6: nat,X_11: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),X_11))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),Y_8))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_11),Y_8)),Z_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_11),Z_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Y_8),Z_6)) ) ) ) ).

tff(fact_2581_div__power,axiom,
    ! [N_27: nat,Y_7: code_code_numeral,X_10: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_7),X_10))
     => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_10),Y_7)),N_27) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_10),N_27)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_7),N_27)) ) ) ).

tff(fact_2582_div__power,axiom,
    ! [N_27: nat,Y_7: quickcheck_code_int,X_10: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_7),X_10))
     => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_10),Y_7)),N_27) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_10),N_27)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_7),N_27)) ) ) ).

tff(fact_2583_div__power,axiom,
    ! [N_27: nat,Y_7: int,X_10: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_7),X_10))
     => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_10),Y_7)),N_27) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_10),N_27)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_7),N_27)) ) ) ).

tff(fact_2584_div__power,axiom,
    ! [N_27: nat,Y_7: nat,X_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_7),X_10))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_10),Y_7)),N_27) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_10),N_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_7),N_27)) ) ) ).

tff(fact_2585_split__neg__lemma,axiom,
    ! [P_1: fun_int_fun_int_bool,N_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),K_1)),hAPP_int_int(div_mod_int(N_1),K_1)))
      <=> ! [I_2: int,J_1: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
              & ( N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ).

tff(fact_2586_split__pos__lemma,axiom,
    ! [P_1: fun_int_fun_int_bool,N_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),K_1)),hAPP_int_int(div_mod_int(N_1),K_1)))
      <=> ! [I_2: int,J_1: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
              & ( N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ).

tff(fact_2587_dvd__eq__mod__eq__0__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_i769753017umeral(number1443263063umeral,X_1)),hAPP_i769753017umeral(number1443263063umeral,Y_2)))
    <=> ( hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_i769753017umeral(number1443263063umeral,Y_2)),hAPP_i769753017umeral(number1443263063umeral,X_1)) = zero_z126310315umeral ) ) ).

tff(fact_2588_dvd__eq__mod__eq__0__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_i1732201573de_int(number1226105091de_int,X_1)),hAPP_i1732201573de_int(number1226105091de_int,Y_2)))
    <=> ( hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_i1732201573de_int(number1226105091de_int,Y_2)),hAPP_i1732201573de_int(number1226105091de_int,X_1)) = zero_z891286103de_int ) ) ).

tff(fact_2589_dvd__eq__mod__eq__0__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_1)) = zero_zero_int ) ) ).

tff(fact_2590_dvd__eq__mod__eq__0__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,X_1)),hAPP_int_nat(number_number_of_nat,Y_2)))
    <=> ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,Y_2)),hAPP_int_nat(number_number_of_nat,X_1)) = zero_zero_nat ) ) ).

tff(fact_2591_StandardRes__ubound,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(standardRes(P_3),X)),P_3)) ) ).

tff(fact_2592_StandardRes__eq__zcong,axiom,
    ! [Ma: int,X_1: int] :
      ( ( hAPP_int_int(standardRes(Ma),X_1) = zero_zero_int )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),Ma)) ) ).

tff(fact_2593_StandardRes__prop3,axiom,
    ! [X_1: int,P_5: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5))
    <=> ( hAPP_int_int(standardRes(P_5),X_1) != zero_zero_int ) ) ).

tff(fact_2594_int__div__less__self,axiom,
    ! [K_2: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),K_2)),X)) ) ) ).

tff(fact_2595_zdiv__mono1__neg,axiom,
    ! [B_2: int,A_3: int,A_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))) ) ) ).

tff(fact_2596_zdiv__mono1,axiom,
    ! [B_2: int,A_3: int,A_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2))) ) ) ).

tff(fact_2597_div__neg__neg__trivial,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = zero_zero_int ) ) ) ).

tff(fact_2598_zdiv__mono2__neg,axiom,
    ! [B_2: int,B_5: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))) ) ) ) ).

tff(fact_2599_div__nonpos__pos__le0,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int)) ) ) ).

tff(fact_2600_neg__imp__zdiv__nonneg__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

tff(fact_2601_div__pos__pos__trivial,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = zero_zero_int ) ) ) ).

tff(fact_2602_div__nonneg__neg__le0,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int)) ) ) ).

tff(fact_2603_zdiv__mono2,axiom,
    ! [B_2: int,B_5: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5))) ) ) ) ).

tff(fact_2604_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [B: int,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

tff(fact_2605_pos__imp__zdiv__pos__iff,axiom,
    ! [I_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1)) ) ) ).

tff(fact_2606_pos__imp__zdiv__nonneg__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

tff(fact_2607_zdiv__eq__0__iff,axiom,
    ! [I_1: int,K_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1) = zero_zero_int )
    <=> ( ( K_1 = zero_zero_int )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I_1))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) )
        | ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int))
          & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) ) ) ) ).

tff(fact_2608_div__prop1,axiom,
    ! [X: int,Y: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y)) ) ) ).

tff(fact_2609_zdiv__zmult2__eq,axiom,
    ! [A_3: int,B_2: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1) ) ) ).

tff(fact_2610_mod__neg__neg__trivial,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = A_3 ) ) ) ).

tff(fact_2611_neg__mod__conj,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))) ) ) ).

tff(fact_2612_neg__mod__sign,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)) ) ).

tff(fact_2613_mod__pos__pos__trivial,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = A_3 ) ) ) ).

tff(fact_2614_pos__mod__conj,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2)))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) ) ) ).

tff(fact_2615_pos__mod__sign,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))) ) ).

tff(fact_2616_zcong__zmod__eq,axiom,
    ! [A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
      <=> ( hAPP_int_int(div_mod_int(A),Ma) = hAPP_int_int(div_mod_int(B),Ma) ) ) ) ).

tff(fact_2617_zmod__zdvd__zmod,axiom,
    ! [A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),B_2))
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_3),B_2)),M) = hAPP_int_int(div_mod_int(A_3),M) ) ) ) ).

tff(fact_2618_zmod__minus1__right,axiom,
    ! [A_3: int] : hAPP_int_int(div_mod_int(A_3),hAPP_int_int(number_number_of_int,min)) = zero_zero_int ).

tff(fact_2619_zdvd__iff__zmod__eq__0__number__of,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2)))
    <=> ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_1)) = zero_zero_int ) ) ).

tff(fact_2620_div__mult__self1,axiom,
    ! [A_18: int,C_5: int,B_14: int] :
      ( ( B_14 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_18),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_5),B_14))),B_14) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_5),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_18),B_14)) ) ) ).

tff(fact_2621_div__mult__self1,axiom,
    ! [A_18: nat,C_5: nat,B_14: nat] :
      ( ( B_14 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_18),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_5),B_14))),B_14) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_18),B_14)) ) ) ).

tff(fact_2622_div__mult__self1,axiom,
    ! [A_18: code_code_numeral,C_5: code_code_numeral,B_14: code_code_numeral] :
      ( ( B_14 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_18),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_5),B_14))),B_14) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_5),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_18),B_14)) ) ) ).

tff(fact_2623_div__mult__self1,axiom,
    ! [A_18: quickcheck_code_int,C_5: quickcheck_code_int,B_14: quickcheck_code_int] :
      ( ( B_14 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_18),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_5),B_14))),B_14) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_18),B_14)) ) ) ).

tff(fact_2624_div__mult__self2,axiom,
    ! [A_17: int,C_4: int,B_13: int] :
      ( ( B_13 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_17),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_13),C_4))),B_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_17),B_13)) ) ) ).

tff(fact_2625_div__mult__self2,axiom,
    ! [A_17: nat,C_4: nat,B_13: nat] :
      ( ( B_13 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_13),C_4))),B_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_17),B_13)) ) ) ).

tff(fact_2626_div__mult__self2,axiom,
    ! [A_17: code_code_numeral,C_4: code_code_numeral,B_13: code_code_numeral] :
      ( ( B_13 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_17),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_13),C_4))),B_13) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_4),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_17),B_13)) ) ) ).

tff(fact_2627_div__mult__self2,axiom,
    ! [A_17: quickcheck_code_int,C_4: quickcheck_code_int,B_13: quickcheck_code_int] :
      ( ( B_13 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_17),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_13),C_4))),B_13) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_17),B_13)) ) ) ).

tff(fact_2628_div__add__self1,axiom,
    ! [A_16: int,B_12: int] :
      ( ( B_12 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_12),A_16)),B_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_16),B_12)),one_one_int) ) ) ).

tff(fact_2629_div__add__self1,axiom,
    ! [A_16: nat,B_12: nat] :
      ( ( B_12 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_12),A_16)),B_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_16),B_12)),one_one_nat) ) ) ).

tff(fact_2630_div__add__self1,axiom,
    ! [A_16: code_code_numeral,B_12: code_code_numeral] :
      ( ( B_12 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_12),A_16)),B_12) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_16),B_12)),one_on1645066479umeral) ) ) ).

tff(fact_2631_div__add__self1,axiom,
    ! [A_16: quickcheck_code_int,B_12: quickcheck_code_int] :
      ( ( B_12 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_12),A_16)),B_12) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_16),B_12)),one_on1684967323de_int) ) ) ).

tff(fact_2632_div__add__self2,axiom,
    ! [A_15: int,B_11: int] :
      ( ( B_11 != zero_zero_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_15),B_11)),B_11) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_15),B_11)),one_one_int) ) ) ).

tff(fact_2633_div__add__self2,axiom,
    ! [A_15: nat,B_11: nat] :
      ( ( B_11 != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_15),B_11)),B_11) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_15),B_11)),one_one_nat) ) ) ).

tff(fact_2634_div__add__self2,axiom,
    ! [A_15: code_code_numeral,B_11: code_code_numeral] :
      ( ( B_11 != zero_z126310315umeral )
     => ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_15),B_11)),B_11) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_15),B_11)),one_on1645066479umeral) ) ) ).

tff(fact_2635_div__add__self2,axiom,
    ! [A_15: quickcheck_code_int,B_11: quickcheck_code_int] :
      ( ( B_11 != zero_z891286103de_int )
     => ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_15),B_11)),B_11) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_15),B_11)),one_on1684967323de_int) ) ) ).

tff(fact_2636_dvd__div__div__eq__mult,axiom,
    ! [D: code_code_numeral,B: code_code_numeral,C: code_code_numeral,A: code_code_numeral] :
      ( ( A != zero_z126310315umeral )
     => ( ( C != zero_z126310315umeral )
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
         => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C),D))
           => ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,D),C) )
            <=> ( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B),C) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A),D) ) ) ) ) ) ) ).

tff(fact_2637_dvd__div__div__eq__mult,axiom,
    ! [D: quickcheck_code_int,B: quickcheck_code_int,C: quickcheck_code_int,A: quickcheck_code_int] :
      ( ( A != zero_z891286103de_int )
     => ( ( C != zero_z891286103de_int )
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
         => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C),D))
           => ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,D),C) )
            <=> ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A),D) ) ) ) ) ) ) ).

tff(fact_2638_dvd__div__div__eq__mult,axiom,
    ! [D: int,B: int,C: int,A: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C),D))
           => ( ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,D),C) )
            <=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

tff(fact_2639_dvd__div__div__eq__mult,axiom,
    ! [D: nat,B: nat,C: nat,A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),D))
           => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,D),C) )
            <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

tff(fact_2640_dvd__div__eq__mult,axiom,
    ! [C: code_code_numeral,B: code_code_numeral,A: code_code_numeral] :
      ( ( A != zero_z126310315umeral )
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B))
       => ( ( hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = C )
        <=> ( B = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C),A) ) ) ) ) ).

tff(fact_2641_dvd__div__eq__mult,axiom,
    ! [C: quickcheck_code_int,B: quickcheck_code_int,A: quickcheck_code_int] :
      ( ( A != zero_z891286103de_int )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B))
       => ( ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) = C )
        <=> ( B = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C),A) ) ) ) ) ).

tff(fact_2642_dvd__div__eq__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( A != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) = C )
        <=> ( B = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

tff(fact_2643_dvd__div__eq__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( A != zero_zero_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) = C )
        <=> ( B = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

tff(fact_2644_zdiv__leq__prop,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))),X)) ) ).

tff(fact_2645_power__preserves__odd,axiom,
    ! [X_1: int,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)),zOdd))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) ) ) ).

tff(fact_2646_div__eq__minus1,axiom,
    ! [B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(number_number_of_int,min) ) ) ).

tff(fact_2647_mod__pos__neg__trivial,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2) ) ) ) ).

tff(fact_2648_StandardRes__lbound,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(standardRes(P_3),X))) ) ).

tff(fact_2649_Euler_Oaux__2,axiom,
    ! [P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_2650_StandardRes__prop2,axiom,
    ! [X1_2: int,X2: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( ( hAPP_int_int(standardRes(Ma),X1_2) = hAPP_int_int(standardRes(Ma),X2) )
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X1_2),X2),Ma)) ) ) ).

tff(fact_2651_zmod__number__of__Bit0,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) ).

tff(fact_2652_div__prop2,axiom,
    ! [X: int,Y: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1)),Z_1)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y)) ) ) ).

tff(fact_2653_divmod__int__rel__div__eq,axiom,
    ! [A_14: int,B_10: int,Y: int,R_7: int] :
      ( ( A_14 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Y)),R_7) )
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_7))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_7),B_10)) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),R_7))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_7),zero_zero_int)) ) ) )
       => ( ( B_10 != zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_14),B_10) = Y ) ) ) ) ).

tff(fact_2654_split__zdiv,axiom,
    ! [P_1: fun_int_bool,N_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),K_1)))
    <=> ( ( ( K_1 = zero_zero_int )
         => hBOOL(hAPP_int_bool(P_1,zero_zero_int)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
                & ( N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,I_2)) ) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
                & ( N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ) ).

tff(fact_2655_divmod__int__rel__mod__eq,axiom,
    ! [A_14: int,B_10: int,Q_9: int,Y: int] :
      ( ( A_14 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Q_9)),Y) )
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),B_10)) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),Y))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),zero_zero_int)) ) ) )
       => ( ( B_10 != zero_zero_int )
         => ( hAPP_int_int(div_mod_int(A_14),B_10) = Y ) ) ) ) ).

tff(fact_2656_zmult2__lemma__aux2,axiom,
    ! [Q_2: int,B_2: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),zero_zero_int)) ) ) ) ).

tff(fact_2657_zmult2__lemma__aux1,axiom,
    ! [Q_2: int,B_2: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))) ) ) ) ).

tff(fact_2658_zmult2__lemma__aux4,axiom,
    ! [Q_2: int,B_2: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))) ) ) ) ).

tff(fact_2659_zmult2__lemma__aux3,axiom,
    ! [Q_2: int,B_2: int,R_3: int,C_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))) ) ) ) ).

tff(fact_2660_split__zmod,axiom,
    ! [P_1: fun_int_bool,N_1: int,K_1: int] :
      ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(N_1),K_1)))
    <=> ( ( ( K_1 = zero_zero_int )
         => hBOOL(hAPP_int_bool(P_1,N_1)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1))
                & ( N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,J_1)) ) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int))
         => ! [I_2: int,J_1: int] :
              ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1))
                & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
                & ( N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) ) )
             => hBOOL(hAPP_int_bool(P_1,J_1)) ) ) ) ) ).

tff(fact_2661_neq__one__mod__two,axiom,
    ! [X_1: int] :
      ( ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int )
    <=> ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int ) ) ).

tff(fact_2662_div__pos__neg__trivial,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_2663_zmod__minus1,axiom,
    ! [B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => ( hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int) ) ) ).

tff(fact_2664_zdiv__number__of__Bit1,axiom,
    ! [V_2: int,W: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)) ) ) ) ).

tff(fact_2665_negDivAlg__minus1,axiom,
    ! [B_2: int] : negDivAlg(hAPP_int_int(number_number_of_int,min),B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int)) ).

tff(fact_2666_negDivAlg_Osimps,axiom,
    ! [A_3: int,B_2: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
       => ( negDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) ) )
      & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
       => ( negDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ) ).

tff(fact_2667_zprime__zOdd__eq__grt__2,axiom,
    ! [P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) ) ) ).

tff(fact_2668_negDivAlg__eqn__number__of,axiom,
    ! [W: int,V_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))))
         => ( negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))))
         => ( negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) ) ) ) ) ).

tff(fact_2669_negDivAlg__eqn,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
         => ( negDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))
         => ( negDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ) ) ).

tff(fact_2670_neg__zdiv__mult__2,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3) ) ) ).

tff(fact_2671_zOddE,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ~ ! [K: int] : X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K)),one_one_int) ) ).

tff(fact_2672_SRStar__mult__prop2,axiom,
    ! [X_1: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(multInv(P_5),X_1)))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ) ).

tff(fact_2673_mod__div__decomp,axiom,
    ! [A_13: int,B_9: int] : A_13 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_13),B_9)),B_9)),hAPP_int_int(div_mod_int(A_13),B_9)) ).

tff(fact_2674_mod__div__decomp,axiom,
    ! [A_13: nat,B_9: nat] : A_13 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_13),B_9)),B_9)),hAPP_nat_nat(div_mod_nat(A_13),B_9)) ).

tff(fact_2675_mod__div__decomp,axiom,
    ! [A_13: code_code_numeral,B_9: code_code_numeral] : A_13 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_13),B_9)),B_9)),hAPP_C498520661umeral(div_mo1740067990umeral(A_13),B_9)) ).

tff(fact_2676_mod__div__decomp,axiom,
    ! [A_13: quickcheck_code_int,B_9: quickcheck_code_int] : A_13 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_13),B_9)),B_9)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_13),B_9)) ).

tff(fact_2677_zfact__prop,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ) ) ).

tff(fact_2678_Euler__part1,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_int_int(number_number_of_int,min)),P_3)) ) ) ) ) ).

tff(fact_2679_posDivAlg_Osimps,axiom,
    ! [A_3: int,B_2: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
          | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
       => ( posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) ) )
      & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) )
       => ( posDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ) ).

tff(fact_2680_posDivAlg__eqn__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W)))
         => ( posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W)))
         => ( posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),posDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) ) ) ) ) ).

tff(fact_2681_Euler__part3,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int),P_3)) ) ) ) ) ).

tff(fact_2682_div__add1__eq,axiom,
    ! [A_3: nat,B_2: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) ).

tff(fact_2683_nat__if__cong,axiom,
    ! [X_1: int,Y_2: int,P_1: bool] :
      ( ( hBOOL(P_1)
       => ( hAPP_int_nat(nat_1,X_1) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_1),Y_2)) ) )
      & ( ~ hBOOL(P_1)
       => ( hAPP_int_nat(nat_1,Y_2) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_1),Y_2)) ) ) ) ).

tff(fact_2684_mod__mult2__eq,axiom,
    ! [A_3: nat,B_2: nat,C_1: nat] : hAPP_nat_nat(div_mod_nat(A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1))),hAPP_nat_nat(div_mod_nat(A_3),B_2)) ).

tff(fact_2685_div__mult1__eq,axiom,
    ! [A_3: nat,B_2: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) ).

tff(fact_2686_div__mod__equality_H,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N)) ).

tff(fact_2687_mult__div__cancel,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N)) ).

tff(fact_2688_Divides_Omod__div__equality_H,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N)) ).

tff(fact_2689_div__le__mono,axiom,
    ! [K_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),K_2))) ) ).

tff(fact_2690_div__le__dividend,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M)) ).

tff(fact_2691_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hAPP_nat_nat(div_mod_nat(M),N) = M ) ) ).

tff(fact_2692_div__mult2__eq,axiom,
    ! [A_3: nat,B_2: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1) ).

tff(fact_2693_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),M)) ).

tff(fact_2694_mod__mult__distrib2,axiom,
    ! [K_2: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) ).

tff(fact_2695_mod__mult__distrib,axiom,
    ! [M: nat,N: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M),N)),K_2) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2)) ).

tff(fact_2696_nat__div__distrib,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) ) ) ).

tff(fact_2697_Divides_Otransfer__nat__int__functions_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) ) ) ) ).

tff(fact_2698_nat__mod__distrib,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(X),Y)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) ) ) ) ).

tff(fact_2699_Divides_Otransfer__nat__int__functions_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(X),Y)) ) ) ) ).

tff(fact_2700_div__less,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = zero_zero_nat ) ) ).

tff(fact_2701_nat__mult__div__cancel__disj,axiom,
    ! [M: nat,N: nat,K_2: nat] :
      ( ( ( K_2 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) = zero_zero_nat ) )
      & ( ( K_2 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) ) ) ) ).

tff(fact_2702_StandardRes__SRStar__prop3,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
     => ( hAPP_int_int(standardRes(P_5),X_1) = X_1 ) ) ).

tff(fact_2703_mod__less__divisor,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N)),N)) ) ).

tff(fact_2704_transfer__nat__int__numerals_I1_J,axiom,
    zero_zero_nat = hAPP_int_nat(nat_1,zero_zero_int) ).

tff(fact_2705_nat__0,axiom,
    hAPP_int_nat(nat_1,zero_zero_int) = zero_zero_nat ).

tff(fact_2706_mod__eq__0__iff,axiom,
    ! [Ma: nat,D: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(Ma),D) = zero_zero_nat )
    <=> ? [Q_4: nat] : Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D),Q_4) ) ).

tff(fact_2707_transfer__nat__int__relations_I1_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( ( hAPP_int_nat(nat_1,X_1) = hAPP_int_nat(nat_1,Y_2) )
        <=> ( X_1 = Y_2 ) ) ) ) ).

tff(fact_2708_all__nat,axiom,
    ! [P_1: fun_nat_bool] :
      ( ! [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
    <=> ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat_1,X_2))) ) ) ).

tff(fact_2709_ex__nat,axiom,
    ! [P_1: fun_nat_bool] :
      ( ? [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
    <=> ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat_1,X_2))) ) ) ).

tff(fact_2710_eq__nat__nat__iff,axiom,
    ! [Z_3: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_3))
       => ( ( hAPP_int_nat(nat_1,Z_2) = hAPP_int_nat(nat_1,Z_3) )
        <=> ( Z_2 = Z_3 ) ) ) ) ).

tff(fact_2711_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

tff(fact_2712_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => ( hAPP_nat_nat(div_mod_nat(M),N) = M ) )
      & ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => ( hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).

tff(fact_2713_mod__mult__self3,axiom,
    ! [K_2: nat,N: nat,M: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)),M)),N) = hAPP_nat_nat(div_mod_nat(M),N) ).

tff(fact_2714_nat__number__of__def,axiom,
    ! [V_2: int] : hAPP_int_nat(number_number_of_nat,V_2) = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,V_2)) ).

tff(fact_2715_nat__number__of,axiom,
    ! [W: int] : hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,W)) = hAPP_int_nat(number_number_of_nat,W) ).

tff(fact_2716_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

tff(fact_2717_transfer__nat__int__numerals_I2_J,axiom,
    one_one_nat = hAPP_int_nat(nat_1,one_one_int) ).

tff(fact_2718_SRStar__SR__prop,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sr,P_5))) ) ).

tff(fact_2719_div__le__mono2,axiom,
    ! [K_2: nat,N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),M))) ) ) ).

tff(fact_2720_nat__mult__div__cancel1,axiom,
    ! [M: nat,N: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) ) ) ).

tff(fact_2721_div__mult__self__is__m,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

tff(fact_2722_div__mult__self1__is__m,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

tff(fact_2723_div__less__dividend,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M)) ) ) ).

tff(fact_2724_mod__le__divisor,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),N)) ) ).

tff(fact_2725_nat__le__0,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int))
     => ( hAPP_int_nat(nat_1,Z_1) = zero_zero_nat ) ) ).

tff(fact_2726_nat__0__iff,axiom,
    ! [I_1: int] :
      ( ( hAPP_int_nat(nat_1,I_1) = zero_zero_nat )
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) ) ).

tff(fact_2727_zless__nat__conj,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).

tff(fact_2728_nat__mono__iff,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).

tff(fact_2729_transfer__nat__int__relations_I3_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat_1,X_1)),hAPP_int_nat(nat_1,Y_2)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ) ) ).

tff(fact_2730_posDivAlg__0,axiom,
    ! [B_2: int] : posDivAlg(zero_zero_int,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ).

tff(fact_2731_SRStar__def,axiom,
    ! [P_5: int] : hAPP_i1948725293t_bool(sRStar,P_5) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) ).

tff(fact_2732_StandardRes__SRStar__prop1a,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
     => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5)) ) ).

tff(fact_2733_split__div,axiom,
    ! [P_1: fun_nat_bool,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),K_1)))
    <=> ( ( ( K_1 = zero_zero_nat )
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
        & ( ( K_1 != zero_zero_nat )
         => ! [I_2: nat,J_1: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
             => ( ( N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) )
               => hBOOL(hAPP_nat_bool(P_1,I_2)) ) ) ) ) ) ).

tff(fact_2734_zero__less__nat__eq,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat_1,Z_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).

tff(fact_2735_split__mod,axiom,
    ! [P_1: fun_nat_bool,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(N_1),K_1)))
    <=> ( ( ( K_1 = zero_zero_nat )
         => hBOOL(hAPP_nat_bool(P_1,N_1)) )
        & ( ( K_1 != zero_zero_nat )
         => ! [I_2: nat,J_1: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1))
             => ( ( N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) )
               => hBOOL(hAPP_nat_bool(P_1,J_1)) ) ) ) ) ) ).

tff(fact_2736_mod__lemma,axiom,
    ! [Q_2: nat,R_3: nat,B_2: nat,C_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),B_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))) ) ) ).

tff(fact_2737_transfer__nat__int__numerals_I4_J,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_2738_transfer__nat__int__relations_I2_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,X_1)),hAPP_int_nat(nat_1,Y_2)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ) ) ).

tff(fact_2739_nat__less__eq__zless,axiom,
    ! [Z_2: int,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ) ).

tff(fact_2740_nat__le__eq__zle,axiom,
    ! [Z_2: int,W_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),W_1))
        | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat_1,W_1)),hAPP_int_nat(nat_1,Z_2)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ) ).

tff(fact_2741_nat__mult__distrib,axiom,
    ! [Z_4: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,Z_1)),hAPP_int_nat(nat_1,Z_4)) ) ) ).

tff(fact_2742_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) ) ) ) ).

tff(fact_2743_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) ) ) ) ).

tff(fact_2744_nat__add__distrib,axiom,
    ! [Z_4: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
       => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,Z_1)),hAPP_int_nat(nat_1,Z_4)) ) ) ) ).

tff(fact_2745_transfer__nat__int__relations_I4_J,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat_1,X_1)),hAPP_int_nat(nat_1,Y_2)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),Y_2)) ) ) ) ).

tff(fact_2746_nat__diff__distrib,axiom,
    ! [Z_1: int,Z_4: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1))
       => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,Z_1)),hAPP_int_nat(nat_1,Z_4)) ) ) ) ).

tff(fact_2747_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
    ! [N: nat,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat_1,X)),N) = hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) ) ) ).

tff(fact_2748_nat__power__eq,axiom,
    ! [N: nat,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat_1,Z_1)),N) ) ) ).

tff(fact_2749_add__self__div__2,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),M)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = M ).

tff(fact_2750_transfer__nat__int__numerals_I3_J,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_2751_Euler_Oaux__1,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3))
     => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat_1,P_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,P_3)),one_one_nat))) ) ) ).

tff(fact_2752_EvenOdd_Oneg__one__odd__power,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,X_1)) = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_2753_Little__Fermat,axiom,
    ! [X: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))),one_one_int),P_3)) ) ) ).

tff(fact_2754_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_2755_mod2__gr__0,axiom,
    ! [Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
    <=> ( hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_nat ) ) ).

tff(fact_2756_MultInv__def,axiom,
    ! [P_3: int,X: int] : hAPP_int_int(multInv(P_3),X) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_2757_posDivAlg__div__mod,axiom,
    ! [L_2: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( posDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2)) ) ) ) ).

tff(fact_2758_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_2759_Int2_Oaux__2,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_2760_Int2_Oaux__1,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,P_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ).

tff(fact_2761_WilsonRuss_Oinv__def,axiom,
    ! [P_3: int,A_3: int] : hAPP_int_int(inv(P_3),A_3) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3) ).

tff(fact_2762_Euler_Oaux____2,axiom,
    ! [P_3: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_2763_StandardRes__SRStar__prop4,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_1)),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ).

tff(fact_2764_StandardRes__SRStar__prop1,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_1)),hAPP_i1948725293t_bool(sRStar,P_5)))
      <=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5)) ) ) ).

tff(fact_2765_SRStar__mult__prop1,axiom,
    ! [Y_2: int,X_1: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),hAPP_i1948725293t_bool(sRStar,P_5)))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ) ).

tff(fact_2766_Euler__part2,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ) ).

tff(fact_2767_StandardRes__SRStar__prop2,axiom,
    ! [X_1: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => ( hBOOL(hAPP_int_bool(zprime,P_5))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5)))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(multInv(P_5),X_1))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ).

tff(fact_2768_Euler__Criterion,axiom,
    ! [A_3: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3))
     => ( hBOOL(hAPP_int_bool(zprime,P_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(legendre(A_3),P_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) ) ) ).

tff(fact_2769_posDivAlg__eqn,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
         => ( posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))
         => ( posDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) ) ) ) ) ).

tff(fact_2770_posDivAlg__eqn__number__of,axiom,
    ! [W: int,V_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))
         => ( posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))
         => ( posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) ) ) ) ) ).

tff(fact_2771_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_nat )
      | ( hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat )
      | ( hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) )
      | ( hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) ) ) ).

tff(fact_2772_nat__aux__def,axiom,
    ! [I: int,N: nat] : hAPP_nat_nat(nat_aux(I),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,I)),N) ).

tff(fact_2773_zmod__eq__0D,axiom,
    ! [M_14: int,D_4: int] :
      ( ( hAPP_int_int(div_mod_int(M_14),D_4) = zero_zero_int )
     => ? [Q_4: int] : M_14 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_4) ) ).

tff(fact_2774_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(X),N) = hAPP_nat_nat(div_mod_nat(Y),N) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y),X))
       => ? [Q_4: nat] : X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_4)) ) ) ).

tff(fact_2775_phi__prime,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hAPP_int_nat(phi,P_3) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)) ) ) ).

tff(fact_2776_negDivAlg_Opsimps,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B)))
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
         => ( negDivAlg(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)) ) )
        & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)))
              | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
         => ( negDivAlg(A,B) = hAPP_P1975530577nt_int(adjust(B),negDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B))) ) ) ) ) ).

tff(fact_2777_transfer__morphism__nat__int,axiom,
    hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(nat_1),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))) ).

tff(fact_2778_transfer__morphismI,axiom,
    ! [F: fun_int_nat,A_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1)) ).

tff(fact_2779_transfer__morphismI,axiom,
    ! [F: fun_nat_int,A_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1)) ).

tff(fact_2780_transfer__morphism__def,axiom,
    ! [F: fun_int_nat,A_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1)) ).

tff(fact_2781_transfer__morphism__def,axiom,
    ! [F: fun_nat_int,A_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1)) ).

tff(fact_2782_eq__diff__eq_H,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),Z_2) )
    <=> ( Y_2 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Z_2) ) ) ).

tff(fact_2783_negDivAlg_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [A_2: int,B_4: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)))
           => ( ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_4)))
                    | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) )
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_2784_posDivAlg_Opsimps,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B)))
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
            | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
         => ( posDivAlg(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A) ) )
        & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
              | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) )
         => ( posDivAlg(A,B) = hAPP_P1975530577nt_int(adjust(B),posDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B))) ) ) ) ) ).

tff(fact_2785_posDivAlg_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [A_2: int,B_4: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)))
           => ( ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_4))
                    | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) )
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_2786_mod__eqD,axiom,
    ! [M: nat,D_3: nat,R_3: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M),D_3) = R_3 )
     => ? [Q_4: nat] : M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_4),D_3)) ) ).

tff(fact_2787_nat__mod__eq__iff,axiom,
    ! [X_1: nat,N_1: nat,Y_2: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(X_1),N_1) = hAPP_nat_nat(div_mod_nat(Y_2),N_1) )
    <=> ? [Q1: nat,Q2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q2)) ) ).

tff(fact_2788_mod__eq__0D,axiom,
    ! [M_14: nat,D_4: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(M_14),D_4) = zero_zero_nat )
     => ? [Q_4: nat] : M_14 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4) ) ).

tff(fact_2789_zmult2__lemma,axiom,
    ! [C_1: int,A_3: int,B_2: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( ( B_2 != zero_zero_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
         => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Q_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)))) ) ) ) ).

tff(fact_2790_unique__remainder,axiom,
    ! [Q_8: int,R_5: int,A_3: int,B_2: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5)))
       => ( ( B_2 != zero_zero_int )
         => ( R_3 = R_5 ) ) ) ) ).

tff(fact_2791_unique__quotient,axiom,
    ! [Q_8: int,R_5: int,A_3: int,B_2: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5)))
       => ( ( B_2 != zero_zero_int )
         => ( Q_2 = Q_8 ) ) ) ) ).

tff(fact_2792_self__remainder,axiom,
    ! [A_3: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( ( A_3 != zero_zero_int )
       => ( R_3 = zero_zero_int ) ) ) ).

tff(fact_2793_divmod__int__rel__0,axiom,
    ! [B_2: int] :
      ( ( B_2 != zero_zero_int )
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int))) ) ).

tff(fact_2794_self__quotient,axiom,
    ! [A_3: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( ( A_3 != zero_zero_int )
       => ( Q_2 = one_one_int ) ) ) ).

tff(fact_2795_divmod__int__rel__div,axiom,
    ! [A_3: int,B_2: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( ( B_2 != zero_zero_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = Q_2 ) ) ) ).

tff(fact_2796_divmod__int__rel__mod,axiom,
    ! [A_3: int,B_2: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( ( B_2 != zero_zero_int )
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = R_3 ) ) ) ).

tff(fact_2797_negDivAlg__correct,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),negDivAlg(A_3,B_2))) ) ) ).

tff(fact_2798_divmod__int__rel__div__mod,axiom,
    ! [A_3: int,B_2: int] :
      ( ( B_2 != zero_zero_int )
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(div_mod_int(A_3),B_2)))) ) ).

tff(fact_2799_posDivAlg__correct,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),posDivAlg(A_3,B_2))) ) ) ).

tff(fact_2800_zadd1__lemma,axiom,
    ! [B_2: int,Bq: int,Br: int,A_3: int,C_1: int,Aq: int,Ar: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aq),Ar)))
     => ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Bq),Br)))
       => ( ( C_1 != zero_zero_int )
         => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aq),Bq)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1)))) ) ) ) ).

tff(fact_2801_divmod__int__relI,axiom,
    ! [A_3: int,B_2: int,Q_2: int,R_3: int] :
      ( ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) )
     => ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) ) ) )
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) ) ) ).

tff(fact_2802_zmult1__lemma,axiom,
    ! [A_3: int,B_2: int,C_1: int,Q_2: int,R_3: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))
     => ( ( C_1 != zero_zero_int )
       => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1)))) ) ) ).

tff(fact_2803_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(nat_tsub(X),Y)) ) ) ) ).

tff(fact_2804_inv__inv__aux,axiom,
    ! [P_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))))) ) ) ).

tff(fact_2805_two__times__odd__div__two__plus__one,axiom,
    ! [X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int) = X ) ) ).

tff(fact_2806_Code__Numeral_Oint__of__code,axiom,
    ! [K_2: code_code_numeral] :
      ( ( ( K_2 = zero_z126310315umeral )
       => ( code_int_of(K_2) = zero_zero_int ) )
      & ( ( K_2 != zero_z126310315umeral )
       => ( ( ( hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral )
           => ( code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) )
          & ( ( hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z126310315umeral )
           => ( code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) ) ) ) ) ) ).

tff(fact_2807_EvenOdd_Oneg__one__even__power,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,X_1)) = one_one_int ) ) ) ).

tff(fact_2808_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat_1,X_9))),hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2809_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,X_9))),hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2810_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat_1,X_9))),hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2811_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat_1,X_9))),hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2812_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat_1,X_9))),hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2813_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat_1,X_9))),hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2814_of__nat__double,axiom,
    ! [X_9: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9))
     => ( hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat_1,X_9))),hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat_1,X_9))) ) ) ).

tff(fact_2815_less__mono__imp__le__mono,axiom,
    ! [I_1: nat,J: nat,F: fun_nat_nat] :
      ( ! [I_2: nat,J_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J))) ) ) ).

tff(fact_2816_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) ) ).

tff(fact_2817_lessI,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N))) ).

tff(fact_2818_zero__less__Suc,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N))) ).

tff(fact_2819_dvd__1__left,axiom,
    ! [K_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2)) ).

tff(fact_2820_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).

tff(fact_2821_code__numeral__zero__minus__one,axiom,
    hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,zero_z126310315umeral),one_on1645066479umeral) = zero_z126310315umeral ).

tff(fact_2822_number__of__int,axiom,
    ! [N_26: nat] : hAPP_int_nat(number_number_of_nat,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_nat(semiri984289939at_nat,N_26) ).

tff(fact_2823_number__of__int,axiom,
    ! [N_26: nat] : hAPP_int_int(number_number_of_int,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_int(semiri1621563631at_int,N_26) ).

tff(fact_2824_number__of__int,axiom,
    ! [N_26: nat] : hAPP_int_real(number267125858f_real,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26) ).

tff(fact_2825_number__of__int,axiom,
    ! [N_26: nat] : hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_complex(semiri2020571505omplex,N_26) ).

tff(fact_2826_number__of__int,axiom,
    ! [N_26: nat] : hAPP_int_rat(number_number_of_rat,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_rat(semiri151668891at_rat,N_26) ).

tff(fact_2827_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

tff(fact_2828_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_n522471361de_int(semiri1424489471de_int,Ma) = hAPP_n522471361de_int(semiri1424489471de_int,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2829_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_n1108039445umeral(semiri1619134803umeral,Ma) = hAPP_n1108039445umeral(semiri1619134803umeral,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2830_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(semiri984289939at_nat,Ma) = hAPP_nat_nat(semiri984289939at_nat,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2831_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2832_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2833_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_complex(semiri2020571505omplex,Ma) = hAPP_nat_complex(semiri2020571505omplex,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2834_of__nat__eq__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_rat(semiri151668891at_rat,Ma) = hAPP_nat_rat(semiri151668891at_rat,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2835_nat_Oinject,axiom,
    ! [Nat_5: nat,Nat_4: nat] :
      ( ( hAPP_nat_nat(suc,Nat_5) = hAPP_nat_nat(suc,Nat_4) )
    <=> ( Nat_5 = Nat_4 ) ) ).

tff(fact_2836_Suc__n__not__n,axiom,
    ! [N: nat] : hAPP_nat_nat(suc,N) != N ).

tff(fact_2837_n__not__Suc__n,axiom,
    ! [N: nat] : N != hAPP_nat_nat(suc,N) ).

tff(fact_2838_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_13)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_13)) ).

tff(fact_2839_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(suc,M_13)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_nat_nat(semiri984289939at_nat,M_13)) ).

tff(fact_2840_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(suc,M_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(semiri132038758t_real,M_13)) ).

tff(fact_2841_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(suc,M_13)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_13)) ).

tff(fact_2842_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(suc,M_13)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_nat_complex(semiri2020571505omplex,M_13)) ).

tff(fact_2843_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(suc,M_13)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_13)) ).

tff(fact_2844_of__nat__Suc,axiom,
    ! [M_13: nat] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(suc,M_13)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_nat_rat(semiri151668891at_rat,M_13)) ).

tff(fact_2845_int__Suc0__eq__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).

tff(fact_2846_zless__iff__Suc__zadd,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
    <=> ? [N_2: nat] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_2847_int__Suc,axiom,
    ! [M: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M)) ).

tff(fact_2848_Suc__neq__Zero,axiom,
    ! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).

tff(fact_2849_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).

tff(fact_2850_nat_Osimps_I3_J,axiom,
    ! [Nat_3: nat] : hAPP_nat_nat(suc,Nat_3) != zero_zero_nat ).

tff(fact_2851_Suc__not__Zero,axiom,
    ! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).

tff(fact_2852_nat_Osimps_I2_J,axiom,
    ! [Nat_2: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat_2) ).

tff(fact_2853_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).

tff(fact_2854_nat__int,axiom,
    ! [N: nat] : hAPP_int_nat(nat_1,hAPP_nat_int(semiri1621563631at_int,N)) = N ).

tff(fact_2855_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2856_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_2857_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => ( M = N ) ) ) ).

tff(fact_2858_less__trans__Suc,axiom,
    ! [K_2: nat,I: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2)) ) ) ).

tff(fact_2859_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( ( hAPP_nat_nat(suc,M) != N )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N)) ) ) ).

tff(fact_2860_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).

tff(fact_2861_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
       => ( M = N ) ) ) ).

tff(fact_2862_not__less__less__Suc__eq,axiom,
    ! [N_1: nat,Ma: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,Ma)))
      <=> ( N_1 = Ma ) ) ) ).

tff(fact_2863_Suc__less__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Ma)),hAPP_nat_nat(suc,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2864_less__Suc__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
        | ( Ma = N_1 ) ) ) ).

tff(fact_2865_not__less__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,Ma))) ) ).

tff(fact_2866_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) ).

tff(fact_2867_add__Suc,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

tff(fact_2868_add__Suc__right,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

tff(fact_2869_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N)) ).

tff(fact_2870_not__less__eq__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),Ma)) ) ).

tff(fact_2871_le__Suc__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(suc,N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))
        | ( Ma = hAPP_nat_nat(suc,N_1) ) ) ) ).

tff(fact_2872_Suc__le__mono,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(suc,Ma)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma)) ) ).

tff(fact_2873_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))) ) ).

tff(fact_2874_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
       => ( M = hAPP_nat_nat(suc,N) ) ) ) ).

tff(fact_2875_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).

tff(fact_2876_Suc__mult__cancel1,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_2877_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).

tff(fact_2878_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(suc,K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K_2) ).

tff(fact_2879_mod__Suc__eq__Suc__mod,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N))),N) ).

tff(fact_2880_exp__mono__eq,axiom,
    ! [X_1: nat,N_1: nat,Y_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N_1)) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_2881_even__zero__int,axiom,
    hBOOL(hAPP_int_bool(even_odd_even_int,zero_zero_int)) ).

tff(fact_2882_odd__one__int,axiom,
    ~ hBOOL(hAPP_int_bool(even_odd_even_int,one_one_int)) ).

tff(fact_2883_Parity_Oeven__product,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)))
    <=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
        | hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ).

tff(fact_2884_even__times__anything,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ).

tff(fact_2885_anything__times__even,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
     => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ).

tff(fact_2886_Parity_Oodd__times__odd,axiom,
    ! [Y: int,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_2887_Parity_Oodd__plus__odd,axiom,
    ! [Y: int,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_2888_odd__plus__even,axiom,
    ! [Y: int,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_2889_Parity_Oeven__plus__odd,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_2890_Parity_Oeven__plus__even,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
       => hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_2891_even__sum,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) )
        | ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ) ).

tff(fact_2892_even__difference,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) )
        | ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
          & ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) ) ) ) ).

tff(fact_2893_odd__pow,axiom,
    ! [N: nat,X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).

tff(fact_2894_one__not__even,axiom,
    ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,one_one_int),zEven)) ).

tff(fact_2895_even__times__either,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zEven)) ) ).

tff(fact_2896_EvenOdd_Oeven__product,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
        | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) ) ) ).

tff(fact_2897_EvenOdd_Oeven__plus__even,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)),zEven)) ) ) ).

tff(fact_2898_even__diff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) ) ) ).

tff(fact_2899_even__minus__even,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven)) ) ) ).

tff(fact_2900_not__odd__impl__even,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_2901_even__odd__disj,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
      | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_2902_odd__iff__not__even,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
    <=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_2903_even__odd__conj,axiom,
    ! [X_1: int] :
      ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
        & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_2904_zero__le__imp__of__nat,axiom,
    ! [M_12: nat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,M_12))) ).

tff(fact_2905_zero__le__imp__of__nat,axiom,
    ! [M_12: nat] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_12))) ).

tff(fact_2906_zero__le__imp__of__nat,axiom,
    ! [M_12: nat] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_12))) ).

tff(fact_2907_zero__le__imp__of__nat,axiom,
    ! [M_12: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_12))) ).

tff(fact_2908_zero__le__imp__of__nat,axiom,
    ! [M_12: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_12))) ).

tff(fact_2909_zero__le__imp__of__nat,axiom,
    ! [M_12: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_12))) ).

tff(fact_2910_of__nat__0__le__iff,axiom,
    ! [N_25: nat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_25))) ).

tff(fact_2911_of__nat__0__le__iff,axiom,
    ! [N_25: nat] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_25))) ).

tff(fact_2912_of__nat__0__le__iff,axiom,
    ! [N_25: nat] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_25))) ).

tff(fact_2913_of__nat__0__le__iff,axiom,
    ! [N_25: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_25))) ).

tff(fact_2914_of__nat__0__le__iff,axiom,
    ! [N_25: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_25))) ).

tff(fact_2915_of__nat__0__le__iff,axiom,
    ! [N_25: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_25))) ).

tff(fact_2916_of__nat__less__0__iff,axiom,
    ! [M_11: nat] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_11)),zero_zero_rat)) ).

tff(fact_2917_of__nat__less__0__iff,axiom,
    ! [M_11: nat] : ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_11)),zero_z891286103de_int)) ).

tff(fact_2918_of__nat__less__0__iff,axiom,
    ! [M_11: nat] : ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_11)),zero_z126310315umeral)) ).

tff(fact_2919_of__nat__less__0__iff,axiom,
    ! [M_11: nat] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_11)),zero_zero_real)) ).

tff(fact_2920_of__nat__less__0__iff,axiom,
    ! [M_11: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_11)),zero_zero_nat)) ).

tff(fact_2921_of__nat__less__0__iff,axiom,
    ! [M_11: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_11)),zero_zero_int)) ).

tff(fact_2922_of__nat__0,axiom,
    hAPP_nat_rat(semiri151668891at_rat,zero_zero_nat) = zero_zero_rat ).

tff(fact_2923_of__nat__0,axiom,
    hAPP_n522471361de_int(semiri1424489471de_int,zero_zero_nat) = zero_z891286103de_int ).

tff(fact_2924_of__nat__0,axiom,
    hAPP_nat_complex(semiri2020571505omplex,zero_zero_nat) = zero_zero_complex ).

tff(fact_2925_of__nat__0,axiom,
    hAPP_n1108039445umeral(semiri1619134803umeral,zero_zero_nat) = zero_z126310315umeral ).

tff(fact_2926_of__nat__0,axiom,
    hAPP_nat_real(semiri132038758t_real,zero_zero_nat) = zero_zero_real ).

tff(fact_2927_of__nat__0,axiom,
    hAPP_nat_nat(semiri984289939at_nat,zero_zero_nat) = zero_zero_nat ).

tff(fact_2928_of__nat__0,axiom,
    hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).

tff(fact_2929_of__nat__less__imp__less,axiom,
    ! [M_10: nat,N_24: nat] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_10)),hAPP_n1108039445umeral(semiri1619134803umeral,N_24)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) ) ).

tff(fact_2930_of__nat__less__imp__less,axiom,
    ! [M_10: nat,N_24: nat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_10)),hAPP_nat_rat(semiri151668891at_rat,N_24)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) ) ).

tff(fact_2931_of__nat__less__imp__less,axiom,
    ! [M_10: nat,N_24: nat] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_10)),hAPP_n522471361de_int(semiri1424489471de_int,N_24)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) ) ).

tff(fact_2932_of__nat__less__imp__less,axiom,
    ! [M_10: nat,N_24: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_24)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) ) ).

tff(fact_2933_of__nat__less__imp__less,axiom,
    ! [M_10: nat,N_24: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_24)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) ) ).

tff(fact_2934_of__nat__less__imp__less,axiom,
    ! [M_10: nat,N_24: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_24)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)) ) ).

tff(fact_2935_less__imp__of__nat__less,axiom,
    ! [M_9: nat,N_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_9)),hAPP_n1108039445umeral(semiri1619134803umeral,N_23))) ) ).

tff(fact_2936_less__imp__of__nat__less,axiom,
    ! [M_9: nat,N_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_9)),hAPP_nat_rat(semiri151668891at_rat,N_23))) ) ).

tff(fact_2937_less__imp__of__nat__less,axiom,
    ! [M_9: nat,N_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_9)),hAPP_n522471361de_int(semiri1424489471de_int,N_23))) ) ).

tff(fact_2938_less__imp__of__nat__less,axiom,
    ! [M_9: nat,N_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_23))) ) ).

tff(fact_2939_less__imp__of__nat__less,axiom,
    ! [M_9: nat,N_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_23))) ) ).

tff(fact_2940_less__imp__of__nat__less,axiom,
    ! [M_9: nat,N_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_23))) ) ).

tff(fact_2941_of__nat__less__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2942_of__nat__less__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2943_of__nat__less__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2944_of__nat__less__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2945_of__nat__less__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2946_of__nat__less__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2947_of__nat__le__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2948_of__nat__le__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2949_of__nat__le__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2950_of__nat__le__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2951_of__nat__le__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2952_of__nat__le__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2953_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_8)),hAPP_n1108039445umeral(semiri1619134803umeral,N_22)) ).

tff(fact_2954_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,M_8)),hAPP_nat_rat(semiri151668891at_rat,N_22)) ).

tff(fact_2955_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_8)),hAPP_n522471361de_int(semiri1424489471de_int,N_22)) ).

tff(fact_2956_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,M_8)),hAPP_nat_complex(semiri2020571505omplex,N_22)) ).

tff(fact_2957_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,M_8)),hAPP_nat_real(semiri132038758t_real,N_22)) ).

tff(fact_2958_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,M_8)),hAPP_nat_nat(semiri984289939at_nat,N_22)) ).

tff(fact_2959_of__nat__add,axiom,
    ! [M_8: nat,N_22: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_8)),hAPP_nat_int(semiri1621563631at_int,N_22)) ).

tff(fact_2960_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_7)),hAPP_n1108039445umeral(semiri1619134803umeral,N_21)) ).

tff(fact_2961_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(semiri151668891at_rat,M_7)),hAPP_nat_rat(semiri151668891at_rat,N_21)) ).

tff(fact_2962_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_7)),hAPP_n522471361de_int(semiri1424489471de_int,N_21)) ).

tff(fact_2963_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(semiri2020571505omplex,M_7)),hAPP_nat_complex(semiri2020571505omplex,N_21)) ).

tff(fact_2964_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(semiri132038758t_real,M_7)),hAPP_nat_real(semiri132038758t_real,N_21)) ).

tff(fact_2965_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(semiri984289939at_nat,M_7)),hAPP_nat_nat(semiri984289939at_nat,N_21)) ).

tff(fact_2966_of__nat__mult,axiom,
    ! [M_7: nat,N_21: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_7)),hAPP_nat_int(semiri1621563631at_int,N_21)) ).

tff(fact_2967_of__nat__1,axiom,
    hAPP_nat_rat(semiri151668891at_rat,one_one_nat) = one_one_rat ).

tff(fact_2968_of__nat__1,axiom,
    hAPP_n522471361de_int(semiri1424489471de_int,one_one_nat) = one_on1684967323de_int ).

tff(fact_2969_of__nat__1,axiom,
    hAPP_nat_complex(semiri2020571505omplex,one_one_nat) = one_one_complex ).

tff(fact_2970_of__nat__1,axiom,
    hAPP_n1108039445umeral(semiri1619134803umeral,one_one_nat) = one_on1645066479umeral ).

tff(fact_2971_of__nat__1,axiom,
    hAPP_nat_real(semiri132038758t_real,one_one_nat) = one_one_real ).

tff(fact_2972_of__nat__1,axiom,
    hAPP_nat_nat(semiri984289939at_nat,one_one_nat) = one_one_nat ).

tff(fact_2973_of__nat__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).

tff(fact_2974_int__0,axiom,
    hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).

tff(fact_2975_int__eq__0__conv,axiom,
    ! [N_1: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,N_1) = zero_zero_int )
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_2976_transfer__int__nat__numerals_I1_J,axiom,
    zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).

tff(fact_2977_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_6)),N_20) ).

tff(fact_2978_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_6)),N_20) ).

tff(fact_2979_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(semiri151668891at_rat,M_6)),N_20) ).

tff(fact_2980_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(semiri2020571505omplex,M_6)),N_20) ).

tff(fact_2981_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(semiri132038758t_real,M_6)),N_20) ).

tff(fact_2982_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(semiri984289939at_nat,M_6)),N_20) ).

tff(fact_2983_of__nat__power,axiom,
    ! [M_6: nat,N_20: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_6)),N_20) ).

tff(fact_2984_int__less__0__conv,axiom,
    ! [K_2: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_2)),zero_zero_int)) ).

tff(fact_2985_zero__zle__int,axiom,
    ! [N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).

tff(fact_2986_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
    ! [Z_1: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1))) ).

tff(fact_2987_transfer__int__nat__quantifiers_I2_J,axiom,
    ! [P_1: fun_int_bool] :
      ( ? [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(P_1,X_2)) )
    <=> ? [X_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_2988_transfer__int__nat__quantifiers_I1_J,axiom,
    ! [P_1: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_int_bool(P_1,X_2)) )
    <=> ! [X_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

tff(fact_2989_zless__int,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_2990_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ).

tff(fact_2991_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_nat_nat(suc,N_19)) = zero_zero_rat ).

tff(fact_2992_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_nat_nat(suc,N_19)) = zero_z891286103de_int ).

tff(fact_2993_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_nat_nat(suc,N_19)) = zero_zero_complex ).

tff(fact_2994_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_nat_nat(suc,N_19)) = zero_z126310315umeral ).

tff(fact_2995_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_nat_nat(suc,N_19)) = zero_zero_real ).

tff(fact_2996_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_nat_nat(suc,N_19)) = zero_zero_nat ).

tff(fact_2997_power__0__Suc,axiom,
    ! [N_19: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_nat_nat(suc,N_19)) = zero_zero_int ).

tff(fact_2998_zle__int,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_2999_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ).

tff(fact_3000_int__mult,axiom,
    ! [M: nat,N: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) ).

tff(fact_3001_zmult__int,axiom,
    ! [M: nat,N: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).

tff(fact_3002_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ).

tff(fact_3003_zle__iff__zadd,axiom,
    ! [W_1: int,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))
    <=> ? [N_2: nat] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_2)) ) ).

tff(fact_3004_power__Suc2,axiom,
    ! [A_12: quickcheck_code_int,N_18: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_12),N_18)),A_12) ).

tff(fact_3005_power__Suc2,axiom,
    ! [A_12: code_code_numeral,N_18: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_12),N_18)),A_12) ).

tff(fact_3006_power__Suc2,axiom,
    ! [A_12: rat,N_18: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_12),N_18)),A_12) ).

tff(fact_3007_power__Suc2,axiom,
    ! [A_12: complex,N_18: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_12),N_18)),A_12) ).

tff(fact_3008_power__Suc2,axiom,
    ! [A_12: real,N_18: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_12),N_18)),A_12) ).

tff(fact_3009_power__Suc2,axiom,
    ! [A_12: nat,N_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_12),N_18)),A_12) ).

tff(fact_3010_power__Suc2,axiom,
    ! [A_12: int,N_18: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_12),N_18)),A_12) ).

tff(fact_3011_power__Suc,axiom,
    ! [A_11: quickcheck_code_int,N_17: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_11),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),N_17)) ).

tff(fact_3012_power__Suc,axiom,
    ! [A_11: code_code_numeral,N_17: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_11),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),N_17)) ).

tff(fact_3013_power__Suc,axiom,
    ! [A_11: rat,N_17: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_11),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),N_17)) ).

tff(fact_3014_power__Suc,axiom,
    ! [A_11: complex,N_17: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_11),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),N_17)) ).

tff(fact_3015_power__Suc,axiom,
    ! [A_11: real,N_17: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_11),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),N_17)) ).

tff(fact_3016_power__Suc,axiom,
    ! [A_11: nat,N_17: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),N_17)) ).

tff(fact_3017_power__Suc,axiom,
    ! [A_11: int,N_17: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_11),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),N_17)) ).

tff(fact_3018_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: quickcheck_code_int,Q_7: nat] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),Q_7)),X_8) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3019_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: code_code_numeral,Q_7: nat] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),Q_7)),X_8) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3020_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: rat,Q_7: nat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),Q_7)),X_8) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3021_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: complex,Q_7: nat] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),Q_7)),X_8) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3022_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: real,Q_7: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),Q_7)),X_8) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3023_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: nat,Q_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),Q_7)),X_8) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3024_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_8: int,Q_7: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),Q_7)),X_8) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),hAPP_nat_nat(suc,Q_7)) ).

tff(fact_3025_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: quickcheck_code_int,Q_6: nat] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_7),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),Q_6)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3026_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: code_code_numeral,Q_6: nat] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_7),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),Q_6)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3027_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: rat,Q_6: nat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_7),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),Q_6)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3028_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: complex,Q_6: nat] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_7),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),Q_6)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3029_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: real,Q_6: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_7),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),Q_6)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3030_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: nat,Q_6: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),Q_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3031_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_7: int,Q_6: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_7),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),Q_6)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),hAPP_nat_nat(suc,Q_6)) ).

tff(fact_3032_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: quickcheck_code_int,Q_5: nat] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_6),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),Q_5)) ).

tff(fact_3033_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: code_code_numeral,Q_5: nat] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_6),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_6),Q_5)) ).

tff(fact_3034_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: rat,Q_5: nat] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_6),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),Q_5)) ).

tff(fact_3035_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: complex,Q_5: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_6),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),Q_5)) ).

tff(fact_3036_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: real,Q_5: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_6),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),Q_5)) ).

tff(fact_3037_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: nat,Q_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),Q_5)) ).

tff(fact_3038_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_6: int,Q_5: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_6),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),Q_5)) ).

tff(fact_3039_zadd__int,axiom,
    ! [M: nat,N: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

tff(fact_3040_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))),Z_1) ).

tff(fact_3041_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ).

tff(fact_3042_int__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).

tff(fact_3043_transfer__int__nat__numerals_I2_J,axiom,
    one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).

tff(fact_3044_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) ) ).

tff(fact_3045_zdvd__int,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))) ) ).

tff(fact_3046_nat__lt__two__imp__zero__or__one,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))
     => ( ( X = zero_zero_nat )
        | ( X = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_3047_less__Suc__eq__0__disj,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N_1)))
    <=> ( ( Ma = zero_zero_nat )
        | ? [J_1: nat] :
            ( ( Ma = hAPP_nat_nat(suc,J_1) )
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N_1)) ) ) ) ).

tff(fact_3048_less__Suc0,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_3049_gr0__conv__Suc,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
    <=> ? [M_1: nat] : N_1 = hAPP_nat_nat(suc,M_1) ) ).

tff(fact_3050_dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> ( Ma = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

tff(fact_3051_add__is__1,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( N_1 = zero_zero_nat ) )
        | ( ( Ma = zero_zero_nat )
          & ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_3052_one__is__add,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1) )
    <=> ( ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( N_1 = zero_zero_nat ) )
        | ( ( Ma = zero_zero_nat )
          & ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_3053_mult__eq__1__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat) )
        & ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_3054_less__iff__Suc__add,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
    <=> ? [K: nat] : N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)) ) ).

tff(fact_3055_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

tff(fact_3056_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

tff(fact_3057_One__nat__def,axiom,
    one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3058_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).

tff(fact_3059_le__less__Suc__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,Ma)))
      <=> ( N_1 = Ma ) ) ) ).

tff(fact_3060_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) ) ).

tff(fact_3061_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).

tff(fact_3062_Suc__le__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_3063_less__Suc__eq__le,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_3064_less__eq__Suc__le,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),Ma)) ) ).

tff(fact_3065_Suc__mult__less__cancel1,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) ) ).

tff(fact_3066_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),hAPP_nat_nat(suc,M))) ).

tff(fact_3067_mult__Suc,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).

tff(fact_3068_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).

tff(fact_3069_Suc__mult__le__cancel1,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) ) ).

tff(fact_3070_Suc__eq__plus1,axiom,
    ! [N: nat] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

tff(fact_3071_Suc__eq__plus1__left,axiom,
    ! [N: nat] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

tff(fact_3072_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

tff(fact_3073_diff__Suc__1,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N)),one_one_nat) = N ).

tff(fact_3074_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

tff(fact_3075_zdiv__int,axiom,
    ! [A_3: nat,B_2: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) ).

tff(fact_3076_Divides_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y)) ).

tff(fact_3077_div__1,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = M ).

tff(fact_3078_mod__1,axiom,
    ! [M: nat] : hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

tff(fact_3079_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) = N )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = zero_zero_nat ) )
      & ( ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) != N )
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) ) ) ) ).

tff(fact_3080_nat__power__eq__Suc__0__iff,axiom,
    ! [X_1: nat,Ma: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),Ma) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Ma = zero_zero_nat )
        | ( X_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_3081_power__Suc__0,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3082_int__power,axiom,
    ! [M: nat,N: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) ).

tff(fact_3083_zpower__int,axiom,
    ! [M: nat,N: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) ).

tff(fact_3084_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
    ! [X: nat,N: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)) ).

tff(fact_3085_zmod__int,axiom,
    ! [A_3: nat,B_2: nat] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) ).

tff(fact_3086_Divides_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X),Y)) ).

tff(fact_3087_exp__mono__lt,axiom,
    ! [X_1: nat,N_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N_1))))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) ) ).

tff(fact_3088_divides__rexp,axiom,
    ! [N: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_nat_nat(suc,N)))) ) ).

tff(fact_3089_exp__mono__le,axiom,
    ! [X_1: nat,N_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N_1))))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) ) ).

tff(fact_3090_even__power,axiom,
    ! [X_1: int,N_1: nat] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)))
    <=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
        & ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_3091_IntNatAux_Oodd__plus__odd,axiom,
    ! [B: int,A: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zEven)) ) ) ).

tff(fact_3092_IntNatAux_Oeven__plus__odd,axiom,
    ! [B: int,A: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd)) ) ) ).

tff(fact_3093_even__plus__odd__prop1,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zEven)) ) ) ).

tff(fact_3094_even__plus__odd__prop2,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd)) ) ) ).

tff(fact_3095_even__minus__odd,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zOdd)) ) ) ).

tff(fact_3096_odd__minus__even,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zOdd)) ) ) ).

tff(fact_3097_odd__minus__odd,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven)) ) ) ).

tff(fact_3098_of__nat__diff,axiom,
    ! [N_16: nat,M_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
     => ( hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(semiri151668891at_rat,M_5)),hAPP_nat_rat(semiri151668891at_rat,N_16)) ) ) ).

tff(fact_3099_of__nat__diff,axiom,
    ! [N_16: nat,M_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
     => ( hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(semiri2020571505omplex,M_5)),hAPP_nat_complex(semiri2020571505omplex,N_16)) ) ) ).

tff(fact_3100_of__nat__diff,axiom,
    ! [N_16: nat,M_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
     => ( hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_5)),hAPP_nat_real(semiri132038758t_real,N_16)) ) ) ).

tff(fact_3101_of__nat__diff,axiom,
    ! [N_16: nat,M_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_5)),hAPP_nat_int(semiri1621563631at_int,N_16)) ) ) ).

tff(fact_3102_int__le__0__conv,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N_1)),zero_zero_int))
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_3103_power__inject__base,axiom,
    ! [A_10: rat,N_15: nat,B_8: rat] :
      ( ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_8),hAPP_nat_nat(suc,N_15)) )
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_10))
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_8))
         => ( A_10 = B_8 ) ) ) ) ).

tff(fact_3104_power__inject__base,axiom,
    ! [A_10: quickcheck_code_int,N_15: nat,B_8: quickcheck_code_int] :
      ( ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_8),hAPP_nat_nat(suc,N_15)) )
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_10))
       => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_8))
         => ( A_10 = B_8 ) ) ) ) ).

tff(fact_3105_power__inject__base,axiom,
    ! [A_10: code_code_numeral,N_15: nat,B_8: code_code_numeral] :
      ( ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_8),hAPP_nat_nat(suc,N_15)) )
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_10))
       => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_8))
         => ( A_10 = B_8 ) ) ) ) ).

tff(fact_3106_power__inject__base,axiom,
    ! [A_10: real,N_15: nat,B_8: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_8),hAPP_nat_nat(suc,N_15)) )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_10))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_8))
         => ( A_10 = B_8 ) ) ) ) ).

tff(fact_3107_power__inject__base,axiom,
    ! [A_10: nat,N_15: nat,B_8: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_8),hAPP_nat_nat(suc,N_15)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_10))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_8))
         => ( A_10 = B_8 ) ) ) ) ).

tff(fact_3108_power__inject__base,axiom,
    ! [A_10: int,N_15: nat,B_8: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_8),hAPP_nat_nat(suc,N_15)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_10))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_8))
         => ( A_10 = B_8 ) ) ) ) ).

tff(fact_3109_power__le__imp__le__base,axiom,
    ! [A_9: rat,N_14: nat,B_7: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_7),hAPP_nat_nat(suc,N_14))))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_7))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_9),B_7)) ) ) ).

tff(fact_3110_power__le__imp__le__base,axiom,
    ! [A_9: quickcheck_code_int,N_14: nat,B_7: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_9),hAPP_nat_nat(suc,N_14))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_7),hAPP_nat_nat(suc,N_14))))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_7))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_9),B_7)) ) ) ).

tff(fact_3111_power__le__imp__le__base,axiom,
    ! [A_9: code_code_numeral,N_14: nat,B_7: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_9),hAPP_nat_nat(suc,N_14))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_7),hAPP_nat_nat(suc,N_14))))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_7))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_9),B_7)) ) ) ).

tff(fact_3112_power__le__imp__le__base,axiom,
    ! [A_9: real,N_14: nat,B_7: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_7),hAPP_nat_nat(suc,N_14))))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_7))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_7)) ) ) ).

tff(fact_3113_power__le__imp__le__base,axiom,
    ! [A_9: nat,N_14: nat,B_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_7),hAPP_nat_nat(suc,N_14))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_7))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_7)) ) ) ).

tff(fact_3114_power__le__imp__le__base,axiom,
    ! [A_9: int,N_14: nat,B_7: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_7),hAPP_nat_nat(suc,N_14))))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_7))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_7)) ) ) ).

tff(fact_3115_power__gt1,axiom,
    ! [N_13: nat,A_8: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_8))
     => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_8),hAPP_nat_nat(suc,N_13)))) ) ).

tff(fact_3116_power__gt1,axiom,
    ! [N_13: nat,A_8: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_8))
     => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_8),hAPP_nat_nat(suc,N_13)))) ) ).

tff(fact_3117_power__gt1,axiom,
    ! [N_13: nat,A_8: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_8))
     => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_8),hAPP_nat_nat(suc,N_13)))) ) ).

tff(fact_3118_power__gt1,axiom,
    ! [N_13: nat,A_8: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_8))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_8),hAPP_nat_nat(suc,N_13)))) ) ).

tff(fact_3119_power__gt1,axiom,
    ! [N_13: nat,A_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_8))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_8),hAPP_nat_nat(suc,N_13)))) ) ).

tff(fact_3120_power__gt1,axiom,
    ! [N_13: nat,A_8: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_8))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_8),hAPP_nat_nat(suc,N_13)))) ) ).

tff(fact_3121_int__nat__eq,axiom,
    ! [Z_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = Z_1 ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = zero_zero_int ) ) ) ).

tff(fact_3122_int__eq__iff,axiom,
    ! [Ma: nat,Z_2: int] :
      ( ( hAPP_nat_int(semiri1621563631at_int,Ma) = Z_2 )
    <=> ( ( Ma = hAPP_int_nat(nat_1,Z_2) )
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) ) ) ).

tff(fact_3123_nat__0__le,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = Z_1 ) ) ).

tff(fact_3124_zless__nat__eq__int__zless,axiom,
    ! [Ma: nat,Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_int_nat(nat_1,Z_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)) ) ).

tff(fact_3125_zdiff__int,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

tff(fact_3126_one__less__mult,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

tff(fact_3127_n__less__n__mult__m,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).

tff(fact_3128_n__less__m__mult__n,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).

tff(fact_3129_mn__eq__m__one,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = M )
       => ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_3130_prod__mn__less__k,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
         => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = K_2 )
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),K_2)) ) ) ) ) ).

tff(fact_3131_one__less__k,axiom,
    ! [M: nat,K_2: nat] :
      ( ( M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2)) ) ) ).

tff(fact_3132_one__less__m,axiom,
    ! [M: nat,K_2: nat] :
      ( ( M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) )
     => ( ( M != hAPP_nat_nat(suc,zero_zero_nat) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) ) ) ).

tff(fact_3133_diff__Suc__less,axiom,
    ! [I: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I))),N)) ) ).

tff(fact_3134_Suc__pred,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))) = N ) ) ).

tff(fact_3135_one__le__mult__iff,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Ma))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) ) ) ).

tff(fact_3136_nat__1,axiom,
    hAPP_int_nat(nat_1,one_one_int) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3137_diff__Suc__diff__eq1,axiom,
    ! [M: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(suc,J_2)) ) ) ).

tff(fact_3138_diff__Suc__diff__eq2,axiom,
    ! [M: nat,K_2: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)) ) ) ).

tff(fact_3139_nat__one__le__power,axiom,
    ! [N: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),I))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))) ) ).

tff(fact_3140_mod__mult__self4,axiom,
    ! [K_2: nat,N: nat,M: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)),M))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) ).

tff(fact_3141_zero__code__numeral__code,axiom,
    zero_z126310315umeral = hAPP_i769753017umeral(number1443263063umeral,pls) ).

tff(fact_3142_of__nat__0__less__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3143_of__nat__0__less__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3144_of__nat__0__less__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3145_of__nat__0__less__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3146_of__nat__0__less__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3147_of__nat__0__less__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3148_power__preserves__even,axiom,
    ! [X_1: int,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)),zEven))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ) ).

tff(fact_3149_zero__less__int__conv,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3150_odd__minus__one__even,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),one_one_int)),zEven)) ) ).

tff(fact_3151_transfer__int__nat__numerals_I4_J,axiom,
    hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3152_zmult__zless__mono2__lemma,axiom,
    ! [K_2: nat,I: int,J_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),J_2))) ) ) ).

tff(fact_3153_realpow__Suc__le__self,axiom,
    ! [N_12: nat,R_6: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),R_6))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,R_6),one_one_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).

tff(fact_3154_realpow__Suc__le__self,axiom,
    ! [N_12: nat,R_6: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),R_6))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,R_6),one_on1684967323de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).

tff(fact_3155_realpow__Suc__le__self,axiom,
    ! [N_12: nat,R_6: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),R_6))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,R_6),one_on1645066479umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).

tff(fact_3156_realpow__Suc__le__self,axiom,
    ! [N_12: nat,R_6: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_6))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_6),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).

tff(fact_3157_realpow__Suc__le__self,axiom,
    ! [N_12: nat,R_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),R_6))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,R_6),one_one_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).

tff(fact_3158_realpow__Suc__le__self,axiom,
    ! [N_12: nat,R_6: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_6))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_6),one_one_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,R_6),hAPP_nat_nat(suc,N_12))),R_6)) ) ) ).

tff(fact_3159_power__Suc__less__one,axiom,
    ! [N_11: nat,A_7: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_7))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_7),one_one_rat))
       => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_7),hAPP_nat_nat(suc,N_11))),one_one_rat)) ) ) ).

tff(fact_3160_power__Suc__less__one,axiom,
    ! [N_11: nat,A_7: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_7))
     => ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_7),one_on1684967323de_int))
       => hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_7),hAPP_nat_nat(suc,N_11))),one_on1684967323de_int)) ) ) ).

tff(fact_3161_power__Suc__less__one,axiom,
    ! [N_11: nat,A_7: code_code_numeral] :
      ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_7))
     => ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_7),one_on1645066479umeral))
       => hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_7),hAPP_nat_nat(suc,N_11))),one_on1645066479umeral)) ) ) ).

tff(fact_3162_power__Suc__less__one,axiom,
    ! [N_11: nat,A_7: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_7))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_7),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_7),hAPP_nat_nat(suc,N_11))),one_one_real)) ) ) ).

tff(fact_3163_power__Suc__less__one,axiom,
    ! [N_11: nat,A_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_7))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_7),one_one_nat))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_7),hAPP_nat_nat(suc,N_11))),one_one_nat)) ) ) ).

tff(fact_3164_power__Suc__less__one,axiom,
    ! [N_11: nat,A_7: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_7))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_7),one_one_int))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_7),hAPP_nat_nat(suc,N_11))),one_one_int)) ) ) ).

tff(fact_3165_split__nat,axiom,
    ! [P_1: fun_nat_bool,I_1: int] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat_1,I_1)))
    <=> ( ! [N_2: nat] :
            ( ( I_1 = hAPP_nat_int(semiri1621563631at_int,N_2) )
           => hBOOL(hAPP_nat_bool(P_1,N_2)) )
        & ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int))
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ).

tff(fact_3166_nat__eq__iff2,axiom,
    ! [Ma: nat,W_1: int] :
      ( ( Ma = hAPP_int_nat(nat_1,W_1) )
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( W_1 = hAPP_nat_int(semiri1621563631at_int,Ma) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( Ma = zero_zero_nat ) ) ) ) ).

tff(fact_3167_nat__eq__iff,axiom,
    ! [W_1: int,Ma: nat] :
      ( ( hAPP_int_nat(nat_1,W_1) = Ma )
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( W_1 = hAPP_nat_int(semiri1621563631at_int,Ma) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
         => ( Ma = zero_zero_nat ) ) ) ) ).

tff(fact_3168_int__eq__iff__number__of,axiom,
    ! [Ma: nat,V: int] :
      ( ( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_int_int(number_number_of_int,V) )
    <=> ( ( Ma = hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,V)) )
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V))) ) ) ).

tff(fact_3169_numeral__3__eq__3,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_3170_numeral__1__eq__Suc__0,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3171_Suc3__eq__add__3,axiom,
    ! [N: nat] : hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N) ).

tff(fact_3172_lemma__realpow__diff,axiom,
    ! [Y_6: quickcheck_code_int,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3173_lemma__realpow__diff,axiom,
    ! [Y_6: code_code_numeral,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3174_lemma__realpow__diff,axiom,
    ! [Y_6: rat,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3175_lemma__realpow__diff,axiom,
    ! [Y_6: complex,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3176_lemma__realpow__diff,axiom,
    ! [Y_6: real,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3177_lemma__realpow__diff,axiom,
    ! [Y_6: nat,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3178_lemma__realpow__diff,axiom,
    ! [Y_6: int,P_6: nat,N_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10))
     => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) ) ) ).

tff(fact_3179_Suc__pred_H,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

tff(fact_3180_Suc__diff__1,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

tff(fact_3181_add__eq__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
      & ( ( M != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).

tff(fact_3182_div__geq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

tff(fact_3183_div__if,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = zero_zero_nat ) )
        & ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ) ).

tff(fact_3184_Suc__times__mod__eq,axiom,
    ! [M: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),K_2))
     => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M))),K_2) = one_one_nat ) ) ).

tff(fact_3185_div__Suc,axiom,
    ! [A_3: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,A_3)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),C_1))),C_1)) ).

tff(fact_3186_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(nat_tsub(X),Y))) ) ) ).

tff(fact_3187_transfer__int__nat__numerals_I3_J,axiom,
    hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3188_tsub__eq,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
     => ( hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) ) ).

tff(fact_3189_nat__less__iff,axiom,
    ! [Ma: nat,W_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,W_1)),Ma))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_nat_int(semiri1621563631at_int,Ma))) ) ) ).

tff(fact_3190_realpow__two__diff,axiom,
    ! [X_5: rat,Y_5: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_5),Y_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_5),Y_5)) ).

tff(fact_3191_realpow__two__diff,axiom,
    ! [X_5: complex,Y_5: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_5),Y_5)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_5),Y_5)) ).

tff(fact_3192_realpow__two__diff,axiom,
    ! [X_5: real,Y_5: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_5),Y_5)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_5),Y_5)) ).

tff(fact_3193_realpow__two__diff,axiom,
    ! [X_5: int,Y_5: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_5),Y_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_5),Y_5)) ).

tff(fact_3194_zdiff__int__split,axiom,
    ! [P_1: fun_int_bool,X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_2))))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1))
         => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2)))) )
        & ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
         => hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).

tff(fact_3195_numeral__2__eq__2,axiom,
    hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_3196_semiring__norm_I115_J,axiom,
    hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_3197_add__2__eq__Suc,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)) ).

tff(fact_3198_add__2__eq__Suc_H,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)) ).

tff(fact_3199_Suc__diff__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,pls)),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3200_one__less__nat__eq,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(nat_1,Z_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Z_2)) ) ).

tff(fact_3201_expand__Suc,axiom,
    ! [V_2: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_2)))
     => ( hAPP_int_nat(number_number_of_nat,V_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)) ) ) ).

tff(fact_3202_div2__Suc__Suc,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3203_mod2__Suc__Suc,axiom,
    ! [M: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3204_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ( hAPP_nat_nat(suc,hAPP_int_nat(nat_1,Z_1)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)) ) ) ).

tff(fact_3205_Suc__div__eq__add3__div__number__of,axiom,
    ! [M: nat,V_2: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2)) ).

tff(fact_3206_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N) ).

tff(fact_3207_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N)) ).

tff(fact_3208_Suc__mod__eq__add3__mod__number__of,axiom,
    ! [M: nat,V_2: int] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2)) ).

tff(fact_3209_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N) ).

tff(fact_3210_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N)) ).

tff(fact_3211_split__div__lemma,axiom,
    ! [Q_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_1)),Ma))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(suc,Q_1)))) )
      <=> ( Q_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N_1) ) ) ) ).

tff(fact_3212_split__div_H,axiom,
    ! [P_1: fun_nat_bool,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N_1)))
    <=> ( ( ( N_1 = zero_zero_nat )
          & hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
        | ? [Q_4: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_4)),Ma))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(suc,Q_4))))
            & hBOOL(hAPP_nat_bool(P_1,Q_4)) ) ) ) ).

tff(fact_3213_le__div__geq,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

tff(fact_3214_of__nat__less__two__power,axiom,
    ! [N_9: nat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,N_9)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9))) ).

tff(fact_3215_of__nat__less__two__power,axiom,
    ! [N_9: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_9)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9))) ).

tff(fact_3216_of__nat__less__two__power,axiom,
    ! [N_9: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_9)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9))) ).

tff(fact_3217_of__nat__number__of__lemma,axiom,
    ! [V_4: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_rat(number_number_of_rat,V_4) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_rat ) ) ) ).

tff(fact_3218_of__nat__number__of__lemma,axiom,
    ! [V_4: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_complex(number528085621omplex,V_4) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_complex ) ) ) ).

tff(fact_3219_of__nat__number__of__lemma,axiom,
    ! [V_4: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_real(number267125858f_real,V_4) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_real ) ) ) ).

tff(fact_3220_of__nat__number__of__lemma,axiom,
    ! [V_4: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_int(number_number_of_int,V_4) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4)))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_int ) ) ) ).

tff(fact_3221_even__equiv__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> ? [Y_1: int] : X_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1) ) ).

tff(fact_3222_int__even__iff__2__dvd,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) ) ).

tff(fact_3223_nat__dvd__iff,axiom,
    ! [Z_2: int,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat_1,Z_2)),Ma))
    <=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma))) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2))
         => ( Ma = zero_zero_nat ) ) ) ) ).

tff(fact_3224_zEvenI,axiom,
    ! [X_1: int,K_1: int] :
      ( ( X_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) ) ).

tff(fact_3225_less__2__cases,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
     => ( ( N = zero_zero_nat )
        | ( N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_3226_nat__2,axiom,
    hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_3227_even__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ) ).

tff(fact_3228_two__times__even__div__two,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X ) ) ).

tff(fact_3229_one__code__numeral__code,axiom,
    one_on1645066479umeral = hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit1,pls)) ).

tff(fact_3230_power__odd__eq,axiom,
    ! [A_6: nat,N_8: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_6),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_8))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_6),N_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3231_power__odd__eq,axiom,
    ! [A_6: int,N_8: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_6),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_8))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_6),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_6),N_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3232_tsub__def,axiom,
    ! [Y: int,X: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => ( hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => ( hAPP_int_int(nat_tsub(X),Y) = zero_zero_int ) ) ) ).

tff(fact_3233_lemma__Suc__Suc__4n__diff__2,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N) ) ) ).

tff(fact_3234_Suc__Suc__mult__two__diff__two,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N) ) ) ).

tff(fact_3235_even__div__2__prop1,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ) ).

tff(fact_3236_even__div__2__l,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3237_int__power__div__base,axiom,
    ! [K_2: int,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),M)),K_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,zero_zero_nat))) ) ) ) ).

tff(fact_3238_even__prod__div__2,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_2) ) ) ).

tff(fact_3239_even__div__2__prop2,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1 ) ) ).

tff(fact_3240_even__sum__div__2,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ).

tff(fact_3241_one__div__nat__number__of,axiom,
    ! [V_3: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,V_3))) ).

tff(fact_3242_neg__one__power__parity,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
       => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
          <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) )
         => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,X_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat_1,Y_2)) ) ) ) ) ).

tff(fact_3243_odd__equiv__def,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
    <=> ? [Y_1: int] : X_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1)),one_one_int) ) ).

tff(fact_3244_odd__plus__one__div__two,axiom,
    ! [X: int] :
      ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) ) ) ).

tff(fact_3245_even__plus__one__div__two,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3246_Suc__mult__two__diff__one,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N) ) ) ).

tff(fact_3247_inv__is__inv__aux,axiom,
    ! [M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M))
     => ( hAPP_nat_nat(suc,hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),one_one_int)) ) ) ).

tff(fact_3248_zEvenE,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ~ ! [K: int] : X_1 != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K) ) ).

tff(fact_3249_mod__induct__0,axiom,
    ! [I_1: nat,P_1: fun_nat_bool,P_5: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5))
         => ( hBOOL(hAPP_nat_bool(P_1,I_2))
           => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5))) ) )
     => ( hBOOL(hAPP_nat_bool(P_1,I_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5))
         => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ).

tff(fact_3250_realpow__pos__nth2,axiom,
    ! [N: nat,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ? [R_1: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
          & ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),hAPP_nat_nat(suc,N)) = A_3 ) ) ) ).

tff(fact_3251_zero__less__imp__eq__int,axiom,
    ! [K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2))
     => ? [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
          & ( K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) ) ) ) ).

tff(fact_3252_mod__induct,axiom,
    ! [J: nat,I_1: nat,P_1: fun_nat_bool,P_5: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5))
         => ( hBOOL(hAPP_nat_bool(P_1,I_2))
           => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5))) ) )
     => ( hBOOL(hAPP_nat_bool(P_1,I_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),P_5))
           => hBOOL(hAPP_nat_bool(P_1,J)) ) ) ) ) ).

tff(fact_3253_dvd_Olift__Suc__mono__le,axiom,
    ! [N_1: nat,N_7: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_7))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,N_7))) ) ) ).

tff(fact_3254_inc__induct,axiom,
    ! [P_1: fun_nat_bool,I_1: nat,J: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(P_1,J))
       => ( ! [I_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J))
             => ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_2)))
               => hBOOL(hAPP_nat_bool(P_1,I_2)) ) )
         => hBOOL(hAPP_nat_bool(P_1,I_1)) ) ) ) ).

tff(fact_3255_int__if__cong,axiom,
    ! [X_1: nat,Y_2: nat,P_1: bool] :
      ( ( hBOOL(P_1)
       => ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_1),Y_2)) ) )
      & ( ~ hBOOL(P_1)
       => ( hAPP_nat_int(semiri1621563631at_int,Y_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_1),Y_2)) ) ) ) ).

tff(fact_3256_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3257_int__int__eq,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,N_1) )
    <=> ( Ma = N_1 ) ) ).

tff(fact_3258_nat__of__aux__code,axiom,
    ! [N: nat,I: code_code_numeral] :
      ( ( ( I = zero_z126310315umeral )
       => ( hAPP_nat_nat(code_nat_of_aux(I),N) = N ) )
      & ( ( I != zero_z126310315umeral )
       => ( hAPP_nat_nat(code_nat_of_aux(I),N) = hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N)) ) ) ) ).

tff(fact_3259_nonneg__eq__int,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1))
     => ~ ! [M_1: nat] : Z_1 != hAPP_nat_int(semiri1621563631at_int,M_1) ) ).

tff(fact_3260_nonneg__int__cases,axiom,
    ! [K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ~ ! [N_2: nat] : K_2 != hAPP_nat_int(semiri1621563631at_int,N_2) ) ).

tff(fact_3261_zero__le__imp__eq__int,axiom,
    ! [K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ? [N_2: nat] : K_2 = hAPP_nat_int(semiri1621563631at_int,N_2) ) ).

tff(fact_3262_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ? [K: nat] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)) ) ).

tff(fact_3263_dvd_Olift__Suc__mono__less,axiom,
    ! [N_1: nat,N_7: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_7))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,N_7)))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_7)),hAPP_nat_nat(F,N_1))) ) ) ) ).

tff(fact_3264_dvd_Olift__Suc__mono__less__iff,axiom,
    ! [N_1: nat,Ma: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) )
     => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,Ma)))
          & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,N_1))) )
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) ) ) ).

tff(fact_3265_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M_1: nat] : N = hAPP_nat_nat(suc,M_1) ) ).

tff(fact_3266_one__mod__nat__number__of,axiom,
    ! [V_3: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_nat_nat(suc,zero_zero_nat) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ).

tff(fact_3267_not__neg__0,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).

tff(fact_3268_not__neg__1,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).

tff(fact_3269_not__neg__int,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N))) ).

tff(fact_3270_neg__def,axiom,
    ! [Z_5: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,Z_5))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),zero_zero_int)) ) ).

tff(fact_3271_not__neg__eq__ge__0,axiom,
    ! [X_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).

tff(fact_3272_neg__number__of__Bit1,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_1))))
    <=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) ) ).

tff(fact_3273_not__neg__number__of__Pls,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,pls))) ).

tff(fact_3274_neg__number__of__Bit0,axiom,
    ! [W_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_1))))
    <=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) ) ).

tff(fact_3275_neg__nat,axiom,
    ! [Z_1: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,Z_1))
     => ( hAPP_int_nat(nat_1,Z_1) = zero_zero_nat ) ) ).

tff(fact_3276_neg__number__of__Min,axiom,
    hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,min))) ).

tff(fact_3277_not__neg__nat,axiom,
    ! [Z_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,Z_1))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,Z_1)) = Z_1 ) ) ).

tff(fact_3278_neg__imp__number__of__eq__0,axiom,
    ! [V_2: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
     => ( hAPP_int_nat(number_number_of_nat,V_2) = zero_zero_nat ) ) ).

tff(fact_3279_eq__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( ( hAPP_int_nat(number_number_of_nat,V) = hAPP_int_nat(number_number_of_nat,V_1) )
    <=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V_1)),zero_zero_int)) )
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
         => ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
             => ( hAPP_int_int(number_number_of_int,V) = zero_zero_int ) )
            & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
             => ( V = V_1 ) ) ) ) ) ) ).

tff(fact_3280_nat__number__of__add__left,axiom,
    ! [V_3: int,K_2: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3))),K_2) ) ) ) ) ) ).

tff(fact_3281_int__nat__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_int ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_int(number_number_of_int,V_2) ) ) ) ).

tff(fact_3282_div__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ).

tff(fact_3283_power__nat__number__of__number__of,axiom,
    ! [W: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W)) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_nat(number_number_of_nat,W))) ) ) ) ).

tff(fact_3284_power__nat__number__of,axiom,
    ! [N: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N) = hAPP_int_nat(nat_1,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),N)) ) ) ) ).

tff(fact_3285_mod__nat__number__of,axiom,
    ! [V_3: int,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3)))
           => ( hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(nat_1,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) ) ) ) ) ) ).

tff(fact_3286_Suc__nat__number__of__add,axiom,
    ! [N: nat,V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))),N) ) ) ) ).

tff(fact_3287_SRStar__card,axiom,
    ! [P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(sRStar,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int) ) ) ).

tff(fact_3288_int__diff__cases,axiom,
    ! [Z_1: int] :
      ~ ! [M_1: nat,N_2: nat] : Z_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)) ).

tff(fact_3289_zero__induct__lemma,axiom,
    ! [I_1: nat,P_1: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,K_1))
     => ( ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))
           => hBOOL(hAPP_nat_bool(P_1,N_2)) )
       => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_1))) ) ) ).

tff(fact_3290_Suc__le__D,axiom,
    ! [N: nat,M_4: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_4))
     => ? [M_1: nat] : M_4 = hAPP_nat_nat(suc,M_1) ) ).

tff(fact_3291_succ__Pls,axiom,
    hAPP_int_int(succ,pls) = hAPP_int_int(bit1,pls) ).

tff(fact_3292_succ__Bit1,axiom,
    ! [K_2: int] : hAPP_int_int(succ,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,hAPP_int_int(succ,K_2)) ).

tff(fact_3293_succ__Bit0,axiom,
    ! [K_2: int] : hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2) ).

tff(fact_3294_succ__def,axiom,
    ! [K_2: int] : hAPP_int_int(succ,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int) ).

tff(fact_3295_succ__Min,axiom,
    hAPP_int_int(succ,min) = pls ).

tff(fact_3296_diff__bin__simps_I2_J,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),min) = hAPP_int_int(succ,K_2) ).

tff(fact_3297_add__Bit1__Bit1,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),hAPP_int_int(succ,L_2))) ).

tff(fact_3298_nat__number__of__add__1,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ) ).

tff(fact_3299_nat__1__add__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ) ).

tff(fact_3300_Suc__nat__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) ) ) ) ).

tff(fact_3301_MultInvPair__card__two,axiom,
    ! [J: int,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
           => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J),zero_zero_int),P_5))
             => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(multInvPair(A,P_5),J)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ) ).

tff(fact_3302_SetS__elems__card,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
           => ! [X_2: fun_int_bool] :
                ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(A,P_5)))
               => ( hAPP_f957591787ol_nat(finite_card_int,X_2) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ) ).

tff(fact_3303_lessE,axiom,
    ! [I: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
     => ( ( K_2 != hAPP_nat_nat(suc,I) )
       => ~ ! [J_1: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
             => ( K_2 != hAPP_nat_nat(suc,J_1) ) ) ) ) ).

tff(fact_3304_Suc__lessE,axiom,
    ! [I: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2))
     => ~ ! [J_1: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
           => ( K_2 != hAPP_nat_nat(suc,J_1) ) ) ) ).

tff(fact_3305_SetS__card,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
           => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(A,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ) ) ) ).

tff(fact_3306_MultInvPair__prop2,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( comple219730294t_bool(setS(A,P_5)) = hAPP_i1948725293t_bool(sRStar,P_5) ) ) ) ) ).

tff(fact_3307_real__lbound__gt__zero,axiom,
    ! [D2: real,D1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2)) ) ) ) ).

tff(fact_3308_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa_2: real,Ya: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).

tff(fact_3309_nat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat_1: nat] : Y != hAPP_nat_nat(suc,Nat_1) ) ).

tff(fact_3310_zero__induct,axiom,
    ! [P_1: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(P_1,K_1))
     => ( ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))
           => hBOOL(hAPP_nat_bool(P_1,N_2)) )
       => hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ).

tff(fact_3311_real__sqrt__eq__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(sqrt,X_1) = hAPP_real_real(sqrt,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3312_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))) ) ).

tff(fact_3313_real__sqrt__le__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ).

tff(fact_3314_real__sqrt__eq__0__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(sqrt,X_1) = zero_zero_real )
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_3315_real__sqrt__zero,axiom,
    hAPP_real_real(sqrt,zero_zero_real) = zero_zero_real ).

tff(fact_3316_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))) ) ).

tff(fact_3317_real__sqrt__less__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ).

tff(fact_3318_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) ).

tff(fact_3319_real__sqrt__power,axiom,
    ! [X: real,K_2: nat] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),K_2) ).

tff(fact_3320_real__sqrt__eq__1__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(sqrt,X_1) = one_one_real )
    <=> ( X_1 = one_one_real ) ) ).

tff(fact_3321_real__sqrt__one,axiom,
    hAPP_real_real(sqrt,one_one_real) = one_one_real ).

tff(fact_3322_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( hAPP_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

tff(fact_3323_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,X))) ) ).

tff(fact_3324_real__sqrt__le__0__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).

tff(fact_3325_real__sqrt__ge__0__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) ) ).

tff(fact_3326_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,X))) ) ).

tff(fact_3327_real__sqrt__not__eq__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(sqrt,X) != zero_zero_real ) ) ).

tff(fact_3328_real__sqrt__lt__0__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ).

tff(fact_3329_real__sqrt__gt__0__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) ) ).

tff(fact_3330_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,X))) ) ).

tff(fact_3331_real__sqrt__le__1__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).

tff(fact_3332_real__sqrt__ge__1__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) ) ).

tff(fact_3333_real__sqrt__lt__1__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ).

tff(fact_3334_real__sqrt__gt__1__iff,axiom,
    ! [Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) ) ).

tff(fact_3335_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y))))) ).

tff(fact_3336_real__sqrt__mult__self__sum__ge__zero,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y))))) ).

tff(fact_3337_real__sqrt__two__ge__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3338_real__sqrt__two__gt__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3339_real__sqrt__two__gt__one,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3340_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( hAPP_real_real(sqrt,X) = Y ) ) ) ).

tff(fact_3341_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X ) ) ).

tff(fact_3342_real__sqrt__pow2__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X_1 )
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) ) ).

tff(fact_3343_real__sqrt__pow2__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3344_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A_3: real,C_1: real,B_2: real,D_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,C_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,D_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).

tff(fact_3345_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3346_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3347_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = Y )
     => ( X = zero_zero_real ) ) ).

tff(fact_3348_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X )
     => ( Y = zero_zero_real ) ) ).

tff(fact_3349_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa_2: real,Ya: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3350_real__sqrt__sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3351_lemma__sqrt__hcomplex__capprox,axiom,
    ! [Y: real,X: real,U_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ) ) ) ).

tff(fact_3352_Union__SetS__finite,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,comple219730294t_bool(setS(A,P_5)))) ) ).

tff(fact_3353_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M_1: nat] : N = hAPP_nat_nat(suc,M_1) ) ).

tff(fact_3354_nat__induct,axiom,
    ! [N_1: nat,P_1: fun_nat_bool] :
      ( hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(P_1,N_2))
           => hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) )
       => hBOOL(hAPP_nat_bool(P_1,N_1)) ) ) ).

tff(fact_3355_real__divide__square__eq,axiom,
    ! [R_3: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),R_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_3) ).

tff(fact_3356_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) ).

tff(fact_3357_wset__fin,axiom,
    ! [A: int,P_5: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(A),P_5))) ).

tff(fact_3358_SetS__elems__finite,axiom,
    ! [A: int,P_5: int,X_2: fun_int_bool] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(A,P_5)))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)) ) ).

tff(fact_3359_d22set__fin,axiom,
    ! [A: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,A))) ).

tff(fact_3360_real__0__le__divide__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))
          | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) )
        & ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
          | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),zero_zero_real)) ) ) ) ).

tff(fact_3361_real__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3362_real__gt__half__sum,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)) ) ).

tff(fact_3363_SRStar__finite,axiom,
    ! [P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(sRStar,P_5))) ) ).

tff(fact_3364_lemma__real__divide__sqrt__less,axiom,
    ! [U_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),U_1)) ) ).

tff(fact_3365_real__average__minus__first,axiom,
    ! [A_3: real,B_2: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3366_real__average__minus__second,axiom,
    ! [B_2: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3367_real__sum__of__halves,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X ).

tff(fact_3368_eq__divide__2__times__iff,axiom,
    ! [X_1: real,Y_2: real,Z_2: real] :
      ( ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_2)) )
    <=> ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),Z_2) ) ) ).

tff(fact_3369_complex__mod__triangle__ineq2,axiom,
    ! [B_2: complex,A_3: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_2),A_3))),hAPP_complex_real(norm_norm_complex,B_2))),hAPP_complex_real(norm_norm_complex,A_3))) ).

tff(fact_3370_SetS__finite,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,setS(A,P_5))) ) ).

tff(fact_3371_le__Suc__ex__iff,axiom,
    ! [K_1: nat,L: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L))
    <=> ? [N_2: nat] : L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_2) ) ).

tff(fact_3372_lemma__MVT,axiom,
    ! [F: fun_real_real,A: real,B: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),A)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),B)) ).

tff(fact_3373_arctan__half,axiom,
    ! [X: real] : hAPP_real_real(arctan,X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ).

tff(fact_3374_lemma__f__mono__add,axiom,
    ! [Ma: nat,No: nat,F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,Ma)),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),No)))) ) ).

tff(fact_3375_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))) ) ).

tff(fact_3376_arctan__zero__zero,axiom,
    hAPP_real_real(arctan,zero_zero_real) = zero_zero_real ).

tff(fact_3377_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))) ) ).

tff(fact_3378_natceiling__add__number__of,axiom,
    ! [X: real,N: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),hAPP_int_nat(number_number_of_nat,N)) ) ) ) ).

tff(fact_3379_real__sqrt__sum__squares__less,axiom,
    ! [Y: real,X: real,U_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ).

tff(fact_3380_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),one_one_real)) ).

tff(fact_3381_natfloor__add__number__of,axiom,
    ! [X: real,N: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),hAPP_int_nat(number_number_of_nat,N)) ) ) ) ).

tff(fact_3382_real__norm__def,axiom,
    ! [R_3: real] : hAPP_real_real(norm_norm_real,R_3) = hAPP_real_real(abs_abs_real,R_3) ).

tff(fact_3383_real__sqrt__abs2,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)) = hAPP_real_real(abs_abs_real,X) ).

tff(fact_3384_natfloor__zero,axiom,
    natfloor(zero_zero_real) = zero_zero_nat ).

tff(fact_3385_zero__le__natfloor,axiom,
    ! [X: real] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X))) ).

tff(fact_3386_natfloor__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))) ) ).

tff(fact_3387_natfloor__number__of__eq,axiom,
    ! [N: int] : natfloor(hAPP_int_real(number267125858f_real,N)) = hAPP_int_nat(number_number_of_nat,N) ).

tff(fact_3388_natfloor__one,axiom,
    natfloor(one_one_real) = one_one_nat ).

tff(fact_3389_natceiling__zero,axiom,
    natceiling(zero_zero_real) = zero_zero_nat ).

tff(fact_3390_zero__le__natceiling,axiom,
    ! [X: real] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X))) ).

tff(fact_3391_natceiling__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),natceiling(Y))) ) ).

tff(fact_3392_natceiling__number__of__eq,axiom,
    ! [N: int] : natceiling(hAPP_int_real(number267125858f_real,N)) = hAPP_int_nat(number_number_of_nat,N) ).

tff(fact_3393_natceiling__one,axiom,
    natceiling(one_one_real) = one_one_nat ).

tff(fact_3394_rabs__ratiotest__lemma,axiom,
    ! [X: real,Y: real,C_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y))))
       => ( X = zero_zero_real ) ) ) ).

tff(fact_3395_abs__add__one__not__less__self,axiom,
    ! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,X)),one_one_real)),X)) ).

tff(fact_3396_sin__bound__lemma,axiom,
    ! [U_1: real,V_2: real,X: real,Y: real] :
      ( ( X = Y )
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_2))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),U_1)),Y))),V_2)) ) ) ).

tff(fact_3397_natfloor__neg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( natfloor(X) = zero_zero_nat ) ) ).

tff(fact_3398_abs__add__one__gt__zero,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(abs_abs_real,X)))) ).

tff(fact_3399_natceiling__neg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( natceiling(X) = zero_zero_nat ) ) ).

tff(fact_3400_le__natfloor__eq__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ).

tff(fact_3401_natceiling__le__eq__one,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),one_one_nat))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ).

tff(fact_3402_real__sqrt__abs,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(abs_abs_real,X) ).

tff(fact_3403_le__mult__natfloor,axiom,
    ! [B_2: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,natfloor(A_3)),natfloor(B_2))),natfloor(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)))) ) ) ).

tff(fact_3404_less__one__imp__sqr__less__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_real)) ) ).

tff(fact_3405_natfloor__add__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),one_one_nat) ) ) ).

tff(fact_3406_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3407_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3408_arctan__add,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y)) = hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))) ) ) ) ).

tff(fact_3409_natceiling__add__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),one_one_nat) ) ) ).

tff(fact_3410_le__natfloor__eq__number__of,axiom,
    ! [X_1: real,N_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,N_1)),natfloor(X_1)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N_1)),X_1)) ) ) ) ).

tff(fact_3411_natceiling__le__eq__number__of,axiom,
    ! [X_1: real,N_1: int] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),hAPP_int_nat(number_number_of_nat,N_1)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(number267125858f_real,N_1))) ) ) ) ).

tff(fact_3412_lemma__interval,axiom,
    ! [B_2: real,A_3: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_2))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [Y_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y_1))),D_2))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),B_2)) ) ) ) ) ) ).

tff(fact_3413_lemma__interval__lt,axiom,
    ! [B_2: real,A_3: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_2))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [Y_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y_1))),D_2))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_2)) ) ) ) ) ) ).

tff(fact_3414_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3415_machin,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))))))))) ).

tff(fact_3416_pi__neq__zero,axiom,
    pi != zero_zero_real ).

tff(fact_3417_abs__int__eq,axiom,
    ! [M: nat] : hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M) ).

tff(fact_3418_zdvd__antisym__abs,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_2),A_3))
       => ( hAPP_int_int(abs_abs_int,A_3) = hAPP_int_int(abs_abs_int,B_2) ) ) ) ).

tff(fact_3419_zdvd__self__abs2,axiom,
    ! [D_3: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,D_3)),D_3)) ).

tff(fact_3420_zdvd__self__abs1,axiom,
    ! [D_3: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),hAPP_int_int(abs_abs_int,D_3))) ).

tff(fact_3421_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( hAPP_int_int(abs_abs_int,M) = one_one_int ) ) ).

tff(fact_3422_pi__ge__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

tff(fact_3423_pi__not__less__zero,axiom,
    ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),zero_zero_real)) ).

tff(fact_3424_pi__gt__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),pi)) ).

tff(fact_3425_zdvd1__eq,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),one_one_int))
    <=> ( hAPP_int_int(abs_abs_int,X_1) = one_one_int ) ) ).

tff(fact_3426_int__nat__abs,axiom,
    ! [X: int] : hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_3427_ln__less__self,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),X)) ) ).

tff(fact_3428_ln__inj__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
       => ( ( hAPP_real_real(ln,X_1) = hAPP_real_real(ln,Y_2) )
        <=> ( X_1 = Y_2 ) ) ) ) ).

tff(fact_3429_ln__less__cancel__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_2)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ) ).

tff(fact_3430_abs__div,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X))
     => ( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) ) ) ).

tff(fact_3431_ln__one,axiom,
    hAPP_real_real(ln,one_one_real) = zero_zero_real ).

tff(fact_3432_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( ( hAPP_real_real(ln,X) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

tff(fact_3433_zabs__less__one__iff,axiom,
    ! [Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_2)),one_one_int))
    <=> ( Z_2 = zero_zero_int ) ) ).

tff(fact_3434_dvd__imp__le__int,axiom,
    ! [D_3: int,I: int] :
      ( ( I != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),I))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,D_3)),hAPP_int_int(abs_abs_int,I))) ) ) ).

tff(fact_3435_zero__le__zpower__abs,axiom,
    ! [X: int,N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X)),N))) ).

tff(fact_3436_nat__abs__mult__distrib,axiom,
    ! [W: int,Z_1: int] : hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,W))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Z_1))) ).

tff(fact_3437_abs__eq__1__iff,axiom,
    ! [Z_2: int] :
      ( ( hAPP_int_int(abs_abs_int,Z_2) = one_one_int )
    <=> ( ( Z_2 = one_one_int )
        | ( Z_2 = hAPP_int_int(number_number_of_int,min) ) ) ) ).

tff(fact_3438_ln__le__cancel__iff,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_2)))
        <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ) ).

tff(fact_3439_ln__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X))) ) ).

tff(fact_3440_ln__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X))) ) ).

tff(fact_3441_ln__gt__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) ) ) ).

tff(fact_3442_ln__eq__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( ( hAPP_real_real(ln,X_1) = zero_zero_real )
      <=> ( X_1 = one_one_real ) ) ) ).

tff(fact_3443_ln__less__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ) ).

tff(fact_3444_ln__less__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),zero_zero_real)) ) ) ).

tff(fact_3445_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) ) ) ).

tff(fact_3446_ln__le__minus__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real))) ) ).

tff(fact_3447_zero__less__zpower__abs__iff,axiom,
    ! [X_1: int,N_1: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),N_1)))
    <=> ( ( X_1 != zero_zero_int )
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_3448_abs__power3__distrib,axiom,
    ! [X: int] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3449_zdvd__mult__cancel1,axiom,
    ! [N_1: int,Ma: int] :
      ( ( Ma != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1)),Ma))
      <=> ( hAPP_int_int(abs_abs_int,N_1) = one_one_int ) ) ) ).

tff(fact_3450_dvd__int__iff,axiom,
    ! [Z_2: int,Ma: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Z_2))),Ma)) ) ).

tff(fact_3451_int__dvd__iff,axiom,
    ! [Ma: nat,Z_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Z_2)))) ) ).

tff(fact_3452_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

tff(fact_3453_ln__ge__zero__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)) ) ) ).

tff(fact_3454_ln__mult,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) ) ) ) ).

tff(fact_3455_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)) ) ).

tff(fact_3456_ln__div,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) ) ) ) ).

tff(fact_3457_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)) ) ).

tff(fact_3458_power2__eq__iff__abs__eq,axiom,
    ! [A: int,B: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
    <=> ( hAPP_int_int(abs_abs_int,A) = hAPP_int_int(abs_abs_int,B) ) ) ).

tff(fact_3459_abs__power2__distrib,axiom,
    ! [A_3: int] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3460_pi__ge__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) ).

tff(fact_3461_pi__less__4,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3462_pi__half__neq__two,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ).

tff(fact_3463_power2__eq1__iff,axiom,
    ! [A_3: int] :
      ( ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int )
     => ( hAPP_int_int(abs_abs_int,A_3) = one_one_int ) ) ).

tff(fact_3464_decr__lemma,axiom,
    ! [X: int,Z_1: int,D_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3))),Z_1)) ) ).

tff(fact_3465_incr__lemma,axiom,
    ! [Z_1: int,X: int,D_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3)))) ) ).

tff(fact_3466_pi__half__le__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3467_pi__half__neq__zero,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_real ).

tff(fact_3468_pi__half__less__two,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3469_pi__half__ge__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3470_pi__half__gt__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3471_arctan__ubound,axiom,
    ! [Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3472_arctan1__eq__pi4,axiom,
    hAPP_real_real(arctan,one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3473_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)))) ) ) ).

tff(fact_3474_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3475_best__odd__division__abs,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
       => ? [N_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_1))))),X_1)) ) ) ).

tff(fact_3476_best__division__abs,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
     => ? [N_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X))))),X)) ) ).

tff(fact_3477_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).

tff(fact_3478_int__val__lemma,axiom,
    ! [K_1: int,F: fun_nat_int,N_1: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1)))
         => ? [I_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
              & ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).

tff(fact_3479_nat0__intermed__int__val,axiom,
    ! [K_1: int,F: fun_nat_int,N_1: nat] :
      ( ! [I_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1)))
         => ? [I_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
              & ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ).

tff(fact_3480_real__sqrt__minus,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X)) ).

tff(fact_3481_arctan__minus,axiom,
    ! [X: real] : hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X)) ).

tff(fact_3482_real__minus__mult__self__le,axiom,
    ! [U_1: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),U_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X))) ).

tff(fact_3483_real__add__minus__iff,axiom,
    ! [X_1: real,A: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X_1 = A ) ) ).

tff(fact_3484_real__add__eq__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real )
    <=> ( Y_2 = hAPP_real_real(uminus_uminus_real,X_1) ) ) ).

tff(fact_3485_abs__le__interval__iff,axiom,
    ! [X_1: real,R_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),R_2))
    <=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,R_2)),X_1))
        & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),R_2)) ) ) ).

tff(fact_3486_real__diff__def,axiom,
    ! [R_3: real,S_1: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),S_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),hAPP_real_real(uminus_uminus_real,S_1)) ).

tff(fact_3487_minus__real__def,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y)) ).

tff(fact_3488_abs__minus__add__cancel,axiom,
    ! [X: real,Y: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y))) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,X))) ).

tff(fact_3489_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X))),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_3490_real__0__le__add__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,X_1)),Y_2)) ) ).

tff(fact_3491_real__add__le__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),hAPP_real_real(uminus_uminus_real,X_1))) ) ).

tff(fact_3492_real__add__less__0__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)),zero_zero_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(uminus_uminus_real,X_1))) ) ).

tff(fact_3493_real__0__less__add__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,X_1)),Y_2)) ) ).

tff(fact_3494_abs__real__def,axiom,
    ! [A_3: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,A_3) = hAPP_real_real(uminus_uminus_real,A_3) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,A_3) = A_3 ) ) ) ).

tff(fact_3495_real__abs__def,axiom,
    ! [R_3: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,R_3) = hAPP_real_real(uminus_uminus_real,R_3) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
       => ( hAPP_real_real(abs_abs_real,R_3) = R_3 ) ) ) ).

tff(fact_3496_abs__sum__triangle__ineq,axiom,
    ! [X: real,Y: real,L_2: real,M: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,L_2)),hAPP_real_real(uminus_uminus_real,M))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,L_2)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,M)))))) ).

tff(fact_3497_realpow__square__minus__le,axiom,
    ! [U_1: real,X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,U_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3498_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))),hAPP_real_real(uminus_uminus_real,X))) ) ) ).

tff(fact_3499_aux5,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
     => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))))) ) ) ).

tff(fact_3500_m2pi__less__pi,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))),pi)) ).

tff(fact_3501_minus__pi__half__less__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) ).

tff(fact_3502_arctan__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y)))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3503_arctan__lbound,axiom,
    ! [Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y))) ).

tff(fact_3504_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X)))) ) ) ).

tff(fact_3505_nat__intermed__int__val,axiom,
    ! [K_1: int,F: fun_nat_int,N_1: nat,Ma: nat] :
      ( ! [I_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1)) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Ma)),K_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1)))
           => ? [I_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))
                & ( hAPP_nat_int(F,I_2) = K_1 ) ) ) ) ) ) ).

tff(fact_3506_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3507_arcsin__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3508_arcsin__lbound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) ) ) ).

tff(fact_3509_arcsin__ubound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3510_negative__zle,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),hAPP_nat_int(semiri1621563631at_int,M))) ).

tff(fact_3511_negative__zless,axiom,
    ! [N: nat,M: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))),hAPP_nat_int(semiri1621563631at_int,M))) ).

tff(fact_3512_minus__Pls,axiom,
    hAPP_int_int(uminus_uminus_int,pls) = pls ).

tff(fact_3513_minus__Bit0,axiom,
    ! [K_2: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit0,hAPP_int_int(uminus_uminus_int,K_2)) ).

tff(fact_3514_zminus__0,axiom,
    hAPP_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

tff(fact_3515_zmult__zminus,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z_1)),W) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W)) ).

tff(fact_3516_minus__numeral__code_I5_J,axiom,
    ! [W: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W)) ).

tff(fact_3517_zminus__zadd__distrib,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),hAPP_int_int(uminus_uminus_int,W)) ).

tff(fact_3518_zminus__zminus,axiom,
    ! [Z_1: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)) = Z_1 ).

tff(fact_3519_zcong__zminus,axiom,
    ! [A: int,B: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),hAPP_int_int(uminus_uminus_int,Ma)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) ) ).

tff(fact_3520_uminus__dvd__conv_I1_J,axiom,
    ! [D: int,Ta: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D)),Ta)) ) ).

tff(fact_3521_uminus__dvd__conv_I2_J,axiom,
    ! [D: int,Ta: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(uminus_uminus_int,Ta))) ) ).

tff(fact_3522_zdiv__zminus2,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) ).

tff(fact_3523_zdiv__zminus__zminus,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) ).

tff(fact_3524_zmod__zminus2,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) ).

tff(fact_3525_zmod__zminus__zminus,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_3),B_2)) ).

tff(fact_3526_zminus__zmod,axiom,
    ! [X: int,M: int] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X)),M) ).

tff(fact_3527_even__neg,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,X_1)))
    <=> hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) ) ).

tff(fact_3528_complex__diff__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),hAPP_complex_complex(uminus473333897omplex,Y)) ).

tff(fact_3529_zadd__zminus__inverse2,axiom,
    ! [Z_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),Z_1) = zero_zero_int ).

tff(fact_3530_negative__eq__positive,axiom,
    ! [N_1: nat,Ma: nat] :
      ( ( hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1)) = hAPP_nat_int(semiri1621563631at_int,Ma) )
    <=> ( ( N_1 = zero_zero_nat )
        & ( Ma = zero_zero_nat ) ) ) ).

tff(fact_3531_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M)))) ).

tff(fact_3532_Int_OMin__def,axiom,
    min = hAPP_int_int(uminus_uminus_int,one_one_int) ).

tff(fact_3533_zmod__zminus1__not__zero,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,K_2)),L_2) != zero_zero_int )
     => ( hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int ) ) ).

tff(fact_3534_zmod__zminus2__not__zero,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) != zero_zero_int )
     => ( hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int ) ) ).

tff(fact_3535_mult__Min,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,min),K_2) = hAPP_int_int(uminus_uminus_int,K_2) ).

tff(fact_3536_diff__int__def,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) ).

tff(fact_3537_diff__int__def__symmetric,axiom,
    ! [Z_1: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) ).

tff(fact_3538_minus__Min,axiom,
    hAPP_int_int(uminus_uminus_int,min) = hAPP_int_int(bit1,pls) ).

tff(fact_3539_int__zle__neg,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Ma))))
    <=> ( ( N_1 = zero_zero_nat )
        & ( Ma = zero_zero_nat ) ) ) ).

tff(fact_3540_negative__zle__0,axiom,
    ! [N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),zero_zero_int)) ).

tff(fact_3541_nat__zminus__int,axiom,
    ! [N: nat] : hAPP_int_nat(nat_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))) = zero_zero_nat ).

tff(fact_3542_zabs__def,axiom,
    ! [I: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,I) = hAPP_int_int(uminus_uminus_int,I) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int))
       => ( hAPP_int_int(abs_abs_int,I) = I ) ) ) ).

tff(fact_3543_minus__numeral__code_I6_J,axiom,
    ! [V_2: int,W: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),hAPP_int_int(uminus_uminus_int,W))) ).

tff(fact_3544_zmod__zminus2__eq__if,axiom,
    ! [A_3: int,B_2: int] :
      ( ( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int )
       => ( hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = zero_zero_int ) )
      & ( ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int )
       => ( hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2) ) ) ) ).

tff(fact_3545_zmod__zminus1__eq__if,axiom,
    ! [A_3: int,B_2: int] :
      ( ( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2) = zero_zero_int ) )
      & ( ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int )
       => ( hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2)) ) ) ) ).

tff(fact_3546_zdiv__minus1__right,axiom,
    ! [A_3: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(uminus_uminus_int,A_3) ).

tff(fact_3547_neg__zminus__int,axiom,
    ! [N: nat] : hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N))))) ).

tff(fact_3548_negative__zless__0,axiom,
    ! [N: nat] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))),zero_zero_int)) ).

tff(fact_3549_not__zle__0__negative,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N))))) ).

tff(fact_3550_power3__minus,axiom,
    ! [A_3: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3551_nat__mult__distrib__neg,axiom,
    ! [Z_4: int,Z_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int))
     => ( hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat_1,hAPP_int_int(uminus_uminus_int,Z_1))),hAPP_int_nat(nat_1,hAPP_int_int(uminus_uminus_int,Z_4))) ) ) ).

tff(fact_3552_zdiv__zminus2__eq__if,axiom,
    ! [A_3: int,B_2: int] :
      ( ( B_2 != zero_zero_int )
     => ( ( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) ) )
        & ( ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) ) ) ) ) ).

tff(fact_3553_zdiv__zminus1__eq__if,axiom,
    ! [A_3: int,B_2: int] :
      ( ( B_2 != zero_zero_int )
     => ( ( ( hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) ) )
        & ( ( hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) ) ) ) ) ).

tff(fact_3554_neg__even__power,axiom,
    ! [A: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat_1,X_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat_1,X_1)) ) ) ) ).

tff(fact_3555_neg__odd__power,axiom,
    ! [A: int,X_1: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat_1,X_1)) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat_1,X_1))) ) ) ) ).

tff(fact_3556_zminus1__lemma,axiom,
    ! [A: int,B: int,Q_1: int,R_2: int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A),B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Q_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Q_1)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),R_2))))) ) ).

tff(fact_3557_negD,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),zero_zero_int))
     => ? [N_2: nat] : X = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_3558_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sgn_sgn_real,X)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X)) ) ) ).

tff(fact_3559_cos__arcsin,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => ( hAPP_real_real(cos,hAPP_real_real(arcsin,X)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3560_ln__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),X)) ) ).

tff(fact_3561_cos__minus,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(cos,X) ).

tff(fact_3562_cos__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),one_one_real)) ).

tff(fact_3563_cos__zero,axiom,
    hAPP_real_real(cos,zero_zero_real) = one_one_real ).

tff(fact_3564_cos__arctan__not__zero,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(arctan,X)) != zero_zero_real ).

tff(fact_3565_real__sgn__eq,axiom,
    ! [X: real] : hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(abs_abs_real,X)) ).

tff(fact_3566_cos__monotone__0__pi_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3567_cos__ge__minus__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X))) ).

tff(fact_3568_abs__cos__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X))),one_one_real)) ).

tff(fact_3569_cos__pi,axiom,
    hAPP_real_real(cos,pi) = hAPP_int_real(number267125858f_real,min) ).

tff(fact_3570_cos__periodic__pi,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X)) ).

tff(fact_3571_real__sgn__pos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(sgn_sgn_real,X) = one_one_real ) ) ).

tff(fact_3572_cos__monotone__0__pi,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3573_cos__monotone__minus__pi__0_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))) ) ) ) ).

tff(fact_3574_real__sgn__neg,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
     => ( hAPP_real_real(sgn_sgn_real,X) = hAPP_int_real(number267125858f_real,min) ) ) ).

tff(fact_3575_real__sgn__def,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( hAPP_real_real(sgn_sgn_real,X) = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
           => ( hAPP_real_real(sgn_sgn_real,X) = one_one_real ) )
          & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
           => ( hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

tff(fact_3576_sgn__real__def,axiom,
    ! [A_3: real] :
      ( ( ( A_3 = zero_zero_real )
       => ( hAPP_real_real(sgn_sgn_real,A_3) = zero_zero_real ) )
      & ( ( A_3 != zero_zero_real )
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
           => ( hAPP_real_real(sgn_sgn_real,A_3) = one_one_real ) )
          & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
           => ( hAPP_real_real(sgn_sgn_real,A_3) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

tff(fact_3577_cos__two__neq__zero,axiom,
    hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_real ).

tff(fact_3578_cos__monotone__minus__pi__0,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))) ) ) ) ).

tff(fact_3579_cos__two__le__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).

tff(fact_3580_cos__two__less__zero,axiom,
    hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).

tff(fact_3581_cos__pi__half,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3582_cos__two__pi,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) = one_one_real ).

tff(fact_3583_cos__periodic,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(cos,X) ).

tff(fact_3584_cos__60,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3585_cos__30,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3586_cos__45,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3587_cos__double__less__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),one_one_real)) ) ) ).

tff(fact_3588_cos__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).

tff(fact_3589_cos__3over2__pi,axiom,
    hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),pi)) = zero_zero_real ).

tff(fact_3590_cos__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).

tff(fact_3591_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).

tff(fact_3592_cos__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),pi))
            & ( hAPP_real_real(cos,X_2) = Y )
            & ! [Y_1: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),pi))
                  & ( hAPP_real_real(cos,Y_1) = Y ) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_3593_cos__is__zero,axiom,
    ? [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
      & ( hAPP_real_real(cos,X_2) = zero_zero_real )
      & ! [Y_1: real] :
          ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
            & ( hAPP_real_real(cos,Y_1) = zero_zero_real ) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_3594_tan__double,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) != zero_zero_real )
       => ( hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(tan,X))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3595_cos__arccos__lemma1,axiom,
    ! [X_4: real,Y_4: real] : hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_3596_tan__minus,axiom,
    ! [X: real] : hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X)) ).

tff(fact_3597_tan__zero,axiom,
    hAPP_real_real(tan,zero_zero_real) = zero_zero_real ).

tff(fact_3598_tan__arctan,axiom,
    ! [Y: real] : hAPP_real_real(tan,hAPP_real_real(arctan,Y)) = Y ).

tff(fact_3599_tan__pi,axiom,
    hAPP_real_real(tan,pi) = zero_zero_real ).

tff(fact_3600_tan__periodic__pi,axiom,
    ! [X: real] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(tan,X) ).

tff(fact_3601_tan__periodic__n,axiom,
    ! [X: real,N: int] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N)),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3602_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( hAPP_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I))
           => ( hAPP_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I))
           => ( hAPP_int_int(sgn_sgn_int,I) = hAPP_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

tff(fact_3603_arccos__cos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
       => ( hAPP_real_real(arccos,hAPP_real_real(cos,X)) = X ) ) ) ).

tff(fact_3604_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
     => ( hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ).

tff(fact_3605_tan__60,axiom,
    hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).

tff(fact_3606_arccos__lbound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y))) ) ) ).

tff(fact_3607_cos__arccos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ) ).

tff(fact_3608_arccos__ubound,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) ) ) ).

tff(fact_3609_arccos__cos2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),X))
       => ( hAPP_real_real(arccos,hAPP_real_real(cos,X)) = hAPP_real_real(uminus_uminus_real,X) ) ) ) ).

tff(fact_3610_tan__periodic,axiom,
    ! [X: real] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3611_tan__45,axiom,
    hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = one_one_real ).

tff(fact_3612_tan__add,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,Y) != zero_zero_real )
       => ( ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) != zero_zero_real )
         => ( hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)))) ) ) ) ) ).

tff(fact_3613_lemma__tan__add1,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,Y) != zero_zero_real )
       => ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3614_tan__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(tan,X))) ) ) ).

tff(fact_3615_arccos__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) ) ) ) ).

tff(fact_3616_tan__monotone,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y)),hAPP_real_real(tan,X))) ) ) ) ).

tff(fact_3617_tan__monotone_H,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
            <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_2)),hAPP_real_real(tan,X_1))) ) ) ) ) ) ).

tff(fact_3618_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y)),pi)) ) ) ) ).

tff(fact_3619_tan__inverse,axiom,
    ! [Y: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y)) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)) ).

tff(fact_3620_tan__30,axiom,
    hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_3621_tan__less__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X)),zero_zero_real)) ) ) ).

tff(fact_3622_arctan,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y)))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
      & ( hAPP_real_real(tan,hAPP_real_real(arctan,Y)) = Y ) ) ).

tff(fact_3623_arctan__tan,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hAPP_real_real(arctan,hAPP_real_real(tan,X)) = X ) ) ) ).

tff(fact_3624_arccos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi))
          & ( hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ) ) ).

tff(fact_3625_tan__total__pi4,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Z))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
          & ( hAPP_real_real(tan,Z) = X ) ) ) ).

tff(fact_3626_tan__total__pos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
          & ( hAPP_real_real(tan,X_2) = Y ) ) ) ).

tff(fact_3627_tan__total,axiom,
    ! [Y: real] :
    ? [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
      & ( hAPP_real_real(tan,X_2) = Y )
      & ! [Y_1: real] :
          ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
            & ( hAPP_real_real(tan,Y_1) = Y ) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_3628_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
      & ( hAPP_real_real(tan,X_2) = Y ) ) ).

tff(fact_3629_lemma__tan__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(tan,X_2))) ) ) ).

tff(fact_3630_tan__half,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hAPP_real_real(tan,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),one_one_real)) ) ) ) ).

tff(fact_3631_cos__pi__eq__zero,axiom,
    ! [M: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3632_sin__arccos,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => ( hAPP_real_real(sin,hAPP_real_real(arccos,X)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ).

tff(fact_3633_real__of__nat__ge__zero,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N))) ).

tff(fact_3634_sin__minus,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).

tff(fact_3635_abs__real__of__nat__cancel,axiom,
    ! [X: nat] : hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X) ).

tff(fact_3636_real__of__nat__inject,axiom,
    ! [N_1: nat,Ma: nat] :
      ( ( hAPP_nat_real(real_nat,N_1) = hAPP_nat_real(real_nat,Ma) )
    <=> ( N_1 = Ma ) ) ).

tff(fact_3637_sin__zero,axiom,
    hAPP_real_real(sin,zero_zero_real) = zero_zero_real ).

tff(fact_3638_natfloor__real__of__nat,axiom,
    ! [N: nat] : natfloor(hAPP_nat_real(real_nat,N)) = N ).

tff(fact_3639_natceiling__real__of__nat,axiom,
    ! [N: nat] : natceiling(hAPP_nat_real(real_nat,N)) = N ).

tff(fact_3640_sin__npi2,axiom,
    ! [N: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N))) = zero_zero_real ).

tff(fact_3641_sin__npi,axiom,
    ! [N: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real ).

tff(fact_3642_real__of__nat__def,axiom,
    real_nat = semiri132038758t_real ).

tff(fact_3643_real__eq__of__nat,axiom,
    real_nat = semiri132038758t_real ).

tff(fact_3644_sin__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X)),one_one_real)) ).

tff(fact_3645_real__of__nat__zero__iff,axiom,
    ! [N_1: nat] :
      ( ( hAPP_nat_real(real_nat,N_1) = zero_zero_real )
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_3646_real__of__nat__zero,axiom,
    hAPP_nat_real(real_nat,zero_zero_nat) = zero_zero_real ).

tff(fact_3647_not__real__of__nat__less__zero,axiom,
    ! [N: nat] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),zero_zero_real)) ).

tff(fact_3648_real__of__nat__less__iff,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,Ma)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) ) ).

tff(fact_3649_sin__pi,axiom,
    hAPP_real_real(sin,pi) = zero_zero_real ).

tff(fact_3650_real__of__nat__le__iff,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,Ma)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma)) ) ).

tff(fact_3651_real__of__nat__mult,axiom,
    ! [M: nat,N: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) ).

tff(fact_3652_real__of__nat__add,axiom,
    ! [M: nat,N: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) ).

tff(fact_3653_real__of__nat__1,axiom,
    hAPP_nat_real(real_nat,one_one_nat) = one_one_real ).

tff(fact_3654_power__real__of__nat,axiom,
    ! [M: nat,N: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) ).

tff(fact_3655_real__of__nat__power,axiom,
    ! [M: nat,N: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N) ).

tff(fact_3656_real__natceiling__ge,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X)))) ).

tff(fact_3657_sin__ge__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3658_sin__gt__zero__pi,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),pi))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3659_real__of__nat__le__zero__cancel__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real))
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_3660_sin__ge__minus__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(sin,X))) ).

tff(fact_3661_real__of__nat__Suc__gt__zero,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) ).

tff(fact_3662_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(cos,X) = one_one_real )
     => ( hAPP_real_real(sin,X) = zero_zero_real ) ) ).

tff(fact_3663_abs__sin__le__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X))),one_one_real)) ).

tff(fact_3664_real__of__nat__one,axiom,
    hAPP_nat_real(real_nat,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_real ).

tff(fact_3665_sin__add,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3666_sin__diff2,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(cos,X))) ).

tff(fact_3667_sin__diff,axiom,
    ! [X: real,Y: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3668_real__of__nat__Suc,axiom,
    ! [N: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real) ).

tff(fact_3669_sin__periodic__pi2,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).

tff(fact_3670_sin__periodic__pi,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).

tff(fact_3671_real__of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) ) ) ).

tff(fact_3672_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X)))) ).

tff(fact_3673_tan__def,axiom,
    ! [X: real] : hAPP_real_real(tan,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,X)) ).

tff(fact_3674_real__natfloor__le,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X))),X)) ) ).

tff(fact_3675_le__natfloor,axiom,
    ! [X: nat,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X)),A_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),natfloor(A_3))) ) ).

tff(fact_3676_natceiling__le,axiom,
    ! [X: real,A_3: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A_3)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A_3)) ) ).

tff(fact_3677_sin__2npi,axiom,
    ! [N: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N))),pi)) = zero_zero_real ).

tff(fact_3678_natfloor__power,axiom,
    ! [N: nat,X: real] :
      ( ( X = hAPP_nat_real(real_nat,natfloor(X)) )
     => ( natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X)),N) ) ) ).

tff(fact_3679_real__of__nat__gt__zero__cancel__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_3680_sin__cos__squared__add3,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,X))) = one_one_real ).

tff(fact_3681_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(sin,X) = zero_zero_real )
     => ( hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X)) = one_one_real ) ) ).

tff(fact_3682_cos__diff2,axiom,
    ! [X: real,Y: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) ).

tff(fact_3683_cos__diff,axiom,
    ! [X: real,Y: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3684_cos__add,axiom,
    ! [X: real,Y: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) ).

tff(fact_3685_nat__less__real__le,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real)),hAPP_nat_real(real_nat,Ma))) ) ).

tff(fact_3686_nat__le__real__less,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,Ma)),one_one_real))) ) ).

tff(fact_3687_le__natfloor__eq,axiom,
    ! [A: nat,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),natfloor(X_1)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X_1)) ) ) ).

tff(fact_3688_tan__npi,axiom,
    ! [N: nat] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real ).

tff(fact_3689_natceiling__le__eq,axiom,
    ! [A: nat,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),A))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,A))) ) ) ).

tff(fact_3690_real__natfloor__add__one__gt,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,natfloor(X))),one_one_real))) ).

tff(fact_3691_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi))) = hAPP_real_real(tan,X) ).

tff(fact_3692_natfloor__subtract,axiom,
    ! [A_3: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X))
     => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X)),A_3) ) ) ).

tff(fact_3693_real__natfloor__gt__diff__one,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X)))) ).

tff(fact_3694_natceiling__subtract,axiom,
    ! [A_3: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X))
     => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X)),A_3) ) ) ).

tff(fact_3695_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3696_real__of__nat__div,axiom,
    ! [N: nat,D_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),N))
       => ( hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,D_3)) ) ) ) ).

tff(fact_3697_ln__realpow,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(ln,X)) ) ) ).

tff(fact_3698_less__natfloor,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N)) ) ) ).

tff(fact_3699_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X))))) ).

tff(fact_3700_real__of__nat__number__of,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_real ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
       => ( hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_real(number267125858f_real,V_2) ) ) ) ).

tff(fact_3701_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X)))),one_one_real)) ).

tff(fact_3702_natfloor__add,axiom,
    ! [A_3: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),A_3) ) ) ).

tff(fact_3703_sin__arcsin,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ).

tff(fact_3704_ge__natfloor__plus__one__imp__gt,axiom,
    ! [Z_1: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(Z_1)),one_one_nat)),N))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N))) ) ).

tff(fact_3705_natfloor__eq,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)))
       => ( natfloor(X) = N ) ) ) ).

tff(fact_3706_natceiling__add,axiom,
    ! [A_3: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),A_3) ) ) ).

tff(fact_3707_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_3708_sin__gt__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3709_sin__cos__npi,axiom,
    ! [N: nat] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) ).

tff(fact_3710_sin__double,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(sin,X))),hAPP_real_real(cos,X)) ).

tff(fact_3711_two__realpow__gt,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N))) ).

tff(fact_3712_LIMSEQ__inverse__realpow__zero__lemma,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),X)),one_one_real)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)),N))) ) ).

tff(fact_3713_add__tan__eq,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( ( hAPP_real_real(cos,Y) != zero_zero_real )
       => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).

tff(fact_3714_real__of__nat__div__aux,axiom,
    ! [X: nat,D_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3))
     => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,X)),hAPP_nat_real(real_nat,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X),D_3))),hAPP_nat_real(real_nat,D_3))) ) ) ).

tff(fact_3715_cos__npi,axiom,
    ! [N: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) ).

tff(fact_3716_cos__npi2,axiom,
    ! [N: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) ).

tff(fact_3717_sin__two__pi,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)) = zero_zero_real ).

tff(fact_3718_sin__periodic,axiom,
    ! [X: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(sin,X) ).

tff(fact_3719_sin__pi__half,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).

tff(fact_3720_sin__30,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3721_sin__60,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3722_sin__45,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3723_fact__lemma,axiom,
    ! [N: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N)) ).

tff(fact_3724_natfloor__div__nat,axiom,
    ! [Y: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y))
       => ( natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_nat_real(real_nat,Y))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X)),Y) ) ) ) ).

tff(fact_3725_natceiling__eq,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)))
       => ( natceiling(X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ) ) ) ).

tff(fact_3726_sin__gt__zero2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).

tff(fact_3727_sin__cos__squared__add2__mult,axiom,
    ! [R_3: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3728_sin__cos__squared__add,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).

tff(fact_3729_sin__cos__squared__add2,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).

tff(fact_3730_sin__squared__eq,axiom,
    ! [X: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3731_cos__squared__eq,axiom,
    ! [X: real] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3732_sin__monotone__2pi_H,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) ) ) ) ).

tff(fact_3733_cos__sin__eq,axiom,
    ! [X: real] : hAPP_real_real(cos,X) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) ).

tff(fact_3734_sin__cos__eq,axiom,
    ! [X: real] : hAPP_real_real(sin,X) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) ).

tff(fact_3735_cos__2npi,axiom,
    ! [N: nat] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N))),pi)) = one_one_real ).

tff(fact_3736_sin__less__zero,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sin,X)),zero_zero_real)) ) ) ).

tff(fact_3737_cos__double,axiom,
    ! [X: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_3738_sin__3over2__pi,axiom,
    hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),pi)) = hAPP_real_real(uminus_uminus_real,one_one_real) ).

tff(fact_3739_minus__sin__cos__eq,axiom,
    ! [X: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3740_arcsin__sin,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => ( hAPP_real_real(arcsin,hAPP_real_real(sin,X)) = X ) ) ) ).

tff(fact_3741_sin__cos__add,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3742_sin__cos__minus,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(sin,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).

tff(fact_3743_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
     => ( hAPP_real_real(sin,hAPP_real_real(arccos,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).

tff(fact_3744_arcsin__pi,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),pi))
          & ( hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ) ).

tff(fact_3745_arcsin,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
          & ( hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ) ).

tff(fact_3746_sin__arccos__lemma1,axiom,
    ! [X_4: real,Y_4: real] : hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3747_sin__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
            & ( hAPP_real_real(sin,X_2) = Y )
            & ! [Y_1: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
                  & ( hAPP_real_real(sin,Y_1) = Y ) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_3748_reals__Archimedean6,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
     => ? [N_2: nat] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),one_one_nat))),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,N_2))) ) ) ).

tff(fact_3749_reals__Archimedean4,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ? [N_2: nat] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),Y)),X))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2))),Y))) ) ) ) ).

tff(fact_3750_reals__Archimedean6a,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
     => ? [N_2: nat] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))) ) ) ).

tff(fact_3751_reals__Archimedean3,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y_1: real] :
        ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),X))) ) ).

tff(fact_3752_lemma__STAR__cos,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N)) = zero_zero_real ) ) ).

tff(fact_3753_sin__zero__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(sin,X_1) = zero_zero_real )
    <=> ( ? [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) )
        | ? [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ) ).

tff(fact_3754_cos__zero__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) = zero_zero_real )
    <=> ( ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) )
        | ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ) ) ).

tff(fact_3755_odd__add,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1)))
    <=> ~ ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))
        <=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) ) ) ).

tff(fact_3756_even__add,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))
      <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) ) ) ).

tff(fact_3757_even__sum__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2)))
    <=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) )
        | ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ) ).

tff(fact_3758_even__zero__nat,axiom,
    hBOOL(hAPP_nat_bool(even_odd_even_nat,zero_zero_nat)) ).

tff(fact_3759_even__Suc,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,X_1)))
    <=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) ) ).

tff(fact_3760_odd__1__nat,axiom,
    ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,one_one_nat)) ).

tff(fact_3761_even__product__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_2)))
    <=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
        | hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ).

tff(fact_3762_real__of__nat__fact__not__zero,axiom,
    ! [N: nat] : hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)) != zero_zero_real ).

tff(fact_3763_odd__pos,axiom,
    ! [N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

tff(fact_3764_even__difference__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_2)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
        | ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) )
        | ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
          & ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) ) ) ) ).

tff(fact_3765_Parity_Otransfer__int__nat__relations,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_1)))
    <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) ) ).

tff(fact_3766_even__nat__def,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).

tff(fact_3767_real__of__nat__fact__ge__zero,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))) ).

tff(fact_3768_real__of__nat__fact__gt__zero,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))) ).

tff(fact_3769_odd__nat__equiv__def2,axiom,
    ! [X_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ? [Y_1: nat] : X_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1)) ) ).

tff(fact_3770_even__nat__equiv__def2,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ? [Y_1: nat] : X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1) ) ).

tff(fact_3771_odd__nat__plus__one__div__two,axiom,
    ! [X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) ) ) ).

tff(fact_3772_even__nat__plus__one__div__two,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ) ) ).

tff(fact_3773_odd__nat__mod__two__eq__one,axiom,
    ! [X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

tff(fact_3774_even__nat__mod__two__eq__zero,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ) ).

tff(fact_3775_odd__nat__equiv__def,axiom,
    ! [X_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ( hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

tff(fact_3776_even__nat__equiv__def,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> ( hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ) ).

tff(fact_3777_even__power__nat,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),Y_2)))
    <=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_2)) ) ) ).

tff(fact_3778_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
     => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

tff(fact_3779_even__num__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
      <=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))) ) ) ).

tff(fact_3780_odd__nat__div__two__times__two__plus__one,axiom,
    ! [X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) = X ) ) ).

tff(fact_3781_even__nat__div__two__times__two,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = X ) ) ).

tff(fact_3782_pos__int__even__equiv__nat__even,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => ( hBOOL(hAPP_int_bool(even_odd_even_int,X_1))
      <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat_1,X_1))) ) ) ).

tff(fact_3783_even__dvd,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)) ) ).

tff(fact_3784_nat__even__iff__2__dvd,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) ) ).

tff(fact_3785_even__mult__two__ex,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
    <=> ? [M_1: nat] : N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1) ) ).

tff(fact_3786_even__even__mod__4__iff,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
    <=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3787_odd__Suc__mult__two__ex,axiom,
    ! [N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
    <=> ? [M_1: nat] : N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1)) ) ).

tff(fact_3788_lemma__even__div2,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3789_lemma__STAR__sin,axiom,
    ! [N_1: nat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,hAPP_nat_bool(even_odd_even_nat,N_1)),zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_1)) = zero_zero_real ).

tff(fact_3790_lemma__not__even__div2,axiom,
    ! [N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ).

tff(fact_3791_lemma__odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) )
     => ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3792_lemma__even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat )
     => hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3793_sin__zero__lemma,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( hAPP_real_real(sin,X) = zero_zero_real )
       => ? [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ) ).

tff(fact_3794_cos__zero__lemma,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( hAPP_real_real(cos,X) = zero_zero_real )
       => ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2))
            & ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ) ) ).

tff(fact_3795_odd__square,axiom,
    ! [N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ? [X_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2)),one_one_nat) ) ).

tff(fact_3796_sin__coeff__def,axiom,
    ! [X_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(sin_coeff,X_2) = zero_zero_real ) )
      & ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(sin_coeff,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2))) ) ) ) ).

tff(fact_3797_even__square,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ? [X_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2) ) ).

tff(fact_3798_cos__coeff__def,axiom,
    ! [X_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(cos_coeff,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2))) ) )
      & ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2))
       => ( hAPP_nat_real(cos_coeff,X_2) = zero_zero_real ) ) ) ).

tff(fact_3799_fact__add__num__eq__if2__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(fact_fact_nat,N) ) )
      & ( ( M != zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) ) ) ) ).

tff(fact_3800_fact__add__num__eq__if__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = one_one_nat ) )
      & ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) != zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),one_one_nat))) ) ) ) ).

tff(fact_3801_fact__nonzero__nat,axiom,
    ! [N: nat] : hAPP_nat_nat(fact_fact_nat,N) != zero_zero_nat ).

tff(fact_3802_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N))) ) ).

tff(fact_3803_fact__1__nat,axiom,
    hAPP_nat_nat(fact_fact_nat,one_one_nat) = one_one_nat ).

tff(fact_3804_Fact_Ofact__0__nat,axiom,
    hAPP_nat_nat(fact_fact_nat,zero_zero_nat) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3805_fact__Suc__0__nat,axiom,
    hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_3806_fact__gt__zero__nat,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N))) ).

tff(fact_3807_fact__less__mono__nat,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N))) ) ) ).

tff(fact_3808_Fact_Ofact__Suc,axiom,
    ! [X: nat] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(fact_fact_nat,X)) ).

tff(fact_3809_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M))) ) ).

tff(fact_3810_fact__ge__one__nat,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N))) ).

tff(fact_3811_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(fact_fact_nat,N))) ).

tff(fact_3812_dvd__fact__nat,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(fact_fact_nat,N))) ) ) ).

tff(fact_3813_fact__plus__one__nat,axiom,
    ! [N: nat] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N)) ).

tff(fact_3814_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M)) = zero_zero_nat ) ) ).

tff(fact_3815_fact__num__eq__if__nat,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( hAPP_nat_nat(fact_fact_nat,M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

tff(fact_3816_fact__reduce__nat,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_nat_nat(fact_fact_nat,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).

tff(fact_3817_Ln_Oaux1,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N)))) ) ) ).

tff(fact_3818_aux4,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))) ) ) ).

tff(fact_3819_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y))) ) ).

tff(fact_3820_ln__unique,axiom,
    ! [Y: real,X: real] :
      ( ( hAPP_real_real(exp_real,Y) = X )
     => ( hAPP_real_real(ln,X) = Y ) ) ).

tff(fact_3821_exp__ln__eq,axiom,
    ! [U_1: real,X: real] :
      ( ( hAPP_real_real(exp_real,U_1) = X )
     => ( hAPP_real_real(ln,X) = U_1 ) ) ).

tff(fact_3822_ln__exp,axiom,
    ! [X: real] : hAPP_real_real(ln,hAPP_real_real(exp_real,X)) = X ).

tff(fact_3823_exp__le__cancel__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ).

tff(fact_3824_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) ) ).

tff(fact_3825_exp__less__cancel__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_2)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ).

tff(fact_3826_INVERSE__ZERO,axiom,
    hAPP_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

tff(fact_3827_exp__inj__iff,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_real(exp_real,X_1) = hAPP_real_real(exp_real,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_3828_abs__exp__cancel,axiom,
    ! [X: real] : hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X)) = hAPP_real_real(exp_real,X) ).

tff(fact_3829_real__sqrt__inverse,axiom,
    ! [X: real] : hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) ).

tff(fact_3830_divide__real__def,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),hAPP_real_real(inverse_inverse_real,Y)) ).

tff(fact_3831_real__divide__def,axiom,
    ! [R: real,S_4: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R),S_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R),hAPP_real_real(inverse_inverse_real,S_4)) ).

tff(fact_3832_not__exp__le__zero,axiom,
    ! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),zero_zero_real)) ).

tff(fact_3833_exp__ge__zero,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X))) ).

tff(fact_3834_exp__gt__zero,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X))) ).

tff(fact_3835_not__exp__less__zero,axiom,
    ! [X: real] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),zero_zero_real)) ).

tff(fact_3836_exp__eq__one__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(exp_real,X_1) = one_one_real )
    <=> ( X_1 = zero_zero_real ) ) ).

tff(fact_3837_real__mult__inverse__cancel,axiom,
    ! [Y: real,U_1: real,X1_1: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_1)),U_1))) ) ) ) ).

tff(fact_3838_real__mult__inverse__cancel2,axiom,
    ! [Y: real,U_1: real,X1_1: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(inverse_inverse_real,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_1)))) ) ) ) ).

tff(fact_3839_real__mult__inverse__left,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),X) = one_one_real ) ) ).

tff(fact_3840_one__le__exp__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(exp_real,X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) ) ).

tff(fact_3841_exp__le__one__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ).

tff(fact_3842_exp__gt__one,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X))) ) ).

tff(fact_3843_exp__less__one__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),one_one_real))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ).

tff(fact_3844_one__less__exp__iff,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_3845_exp__ge__add__one__self,axiom,
    ! [X: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))) ).

tff(fact_3846_exp__real__of__nat__mult,axiom,
    ! [N: nat,X: real] : hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),X)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(exp_real,X)),N) ).

tff(fact_3847_exp__ln__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(exp_real,hAPP_real_real(ln,X_1)) = X_1 )
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_3848_exp__ln,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(exp_real,hAPP_real_real(ln,X)) = X ) ) ).

tff(fact_3849_inv__real__of__nat__fact__ge__zero,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) ).

tff(fact_3850_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),X) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) ) ) ).

tff(fact_3851_inv__real__of__nat__fact__gt__zero,axiom,
    ! [N: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) ).

tff(fact_3852_ln__inverse,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X)) ) ) ).

tff(fact_3853_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))) ) ).

tff(fact_3854_ln__x__over__x__mono,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,one_one_real)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,Y)),Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),X))) ) ) ).

tff(fact_3855_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(inverse_inverse_real,X) ) ) ).

tff(fact_3856_exp__bound,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).

tff(fact_3857_tan__sec,axiom,
    ! [X: real] :
      ( ( hAPP_real_real(cos,X) != zero_zero_real )
     => ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).

tff(fact_3858_lemma__exp__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X_2: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),one_one_real)))
          & ( hAPP_real_real(exp_real,X_2) = Y ) ) ) ).

tff(fact_3859_reals__Archimedean,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))),X)) ) ).

tff(fact_3860_log__base__10__eq1,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(ln,X)) ) ) ).

tff(fact_3861_DERIV__arccos,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(deriv_real(arccos,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ) ) ).

tff(fact_3862_DERIV__sin,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(sin,X_1),hAPP_real_real(cos,X_1))) ).

tff(fact_3863_log__one,axiom,
    ! [A_3: real] : hAPP_real_real(log(A_3),one_one_real) = zero_zero_real ).

tff(fact_3864_Log_Olog__def,axiom,
    ! [A_3: real,X: real] : hAPP_real_real(log(A_3),X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,A_3)) ).

tff(fact_3865_DERIV__cos,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(cos,X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)))) ).

tff(fact_3866_DERIV__ln,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(ln,X_1),hAPP_real_real(inverse_inverse_real,X_1))) ) ).

tff(fact_3867_log__eq__one,axiom,
    ! [A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hAPP_real_real(log(A_3),A_3) = one_one_real ) ) ) ).

tff(fact_3868_log__less__cancel__iff,axiom,
    ! [Y_2: real,X_1: real,A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(A),X_1)),hAPP_real_real(log(A),Y_2)))
          <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ) ) ).

tff(fact_3869_log__ln,axiom,
    ! [X: real] : hAPP_real_real(ln,X) = hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X) ).

tff(fact_3870_DERIV__ln__divide,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(ln,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_1))) ) ).

tff(fact_3871_log__le__cancel__iff,axiom,
    ! [Y_2: real,X_1: real,A: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(log(A),X_1)),hAPP_real_real(log(A),Y_2)))
          <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ) ) ).

tff(fact_3872_log__nat__power,axiom,
    ! [B_2: real,N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(log(B_2),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(log(B_2),X)) ) ) ).

tff(fact_3873_log__mult,axiom,
    ! [Y: real,X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
           => ( hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y)) ) ) ) ) ) ).

tff(fact_3874_log__divide,axiom,
    ! [Y: real,X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
           => ( hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y)) ) ) ) ) ) ).

tff(fact_3875_log__inverse,axiom,
    ! [X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hAPP_real_real(log(A_3),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_3),X)) ) ) ) ) ).

tff(fact_3876_log__eq__div__ln__mult__log,axiom,
    ! [X: real,B_2: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_2))
         => ( ( B_2 != one_one_real )
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
             => ( hAPP_real_real(log(A_3),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,B_2)),hAPP_real_real(ln,A_3))),hAPP_real_real(log(B_2),X)) ) ) ) ) ) ) ).

tff(fact_3877_DERIV__real__sqrt,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(sqrt,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).

tff(fact_3878_DERIV__tan,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) != zero_zero_real )
     => hBOOL(hAPP_real_bool(deriv_real(tan,X_1),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_3879_DERIV__arctan,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(arctan,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_3880_DERIV__real__sqrt__generic,axiom,
    ! [D_1: real,X_1: real] :
      ( ( X_1 != zero_zero_real )
     => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
         => ( D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) )
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
           => ( D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) )
         => hBOOL(hAPP_real_bool(deriv_real(sqrt,X_1),D_1)) ) ) ) ).

tff(fact_3881_DERIV__arcsin,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(deriv_real(arcsin,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ) ).

tff(fact_3882_log__base__10__eq2,axiom,
    ! [X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,X)) ) ) ).

tff(fact_3883_DERIV__const__average,axiom,
    ! [V: fun_real_real,K_1: real,A: real,B: real] :
      ( ( A != B )
     => ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(V,X_2),K_1))
       => ( hAPP_real_real(V,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(V,A)),hAPP_real_real(V,B))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ) ).

tff(fact_3884_DERIV__local__min,axiom,
    ! [D: real,F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,Y_1))) )
         => ( L = zero_zero_real ) ) ) ) ).

tff(fact_3885_DERIV__local__max,axiom,
    ! [D: real,F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y_1)),hAPP_real_real(F,X_1))) )
         => ( L = zero_zero_real ) ) ) ) ).

tff(fact_3886_DERIV__local__const,axiom,
    ! [D: real,F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y_1: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D))
             => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_1) ) )
         => ( L = zero_zero_real ) ) ) ) ).

tff(fact_3887_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),H_1))),hAPP_real_real(F,X_1))) ) ) ) ) ) ).

tff(fact_3888_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),H_1)))) ) ) ) ) ) ).

tff(fact_3889_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),H_1))),hAPP_real_real(F,X_1))) ) ) ) ) ) ).

tff(fact_3890_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,X_1: real,L: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,X_1),L))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
       => ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [H_1: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1))
               => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2))
                 => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),H_1)))) ) ) ) ) ) ).

tff(fact_3891_DERIV__const__ratio__const2,axiom,
    ! [F: fun_real_real,K_1: real,A: real,B: real] :
      ( ( A != B )
     => ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(F,X_2),K_1))
       => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)) = K_1 ) ) ) ).

tff(fact_3892_DERIV__const__ratio__const,axiom,
    ! [F: fun_real_real,K_1: real,A: real,B: real] :
      ( ( A != B )
     => ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(F,X_2),K_1))
       => ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),K_1) ) ) ) ).

tff(fact_3893_DERIV__even__real__root,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N_1))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ) ).

tff(fact_3894_DERIV__real__root__generic,axiom,
    ! [D_1: real,X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ( X_1 != zero_zero_real )
       => ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
             => ( D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
             => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
               => ( D_1 = hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
               => ( D_1 = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))) ) )
             => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),D_1)) ) ) ) ) ) ).

tff(fact_3895_real__root__zero,axiom,
    ! [N: nat] : hAPP_real_real(hAPP_n546711566l_real(root,N),zero_zero_real) = zero_zero_real ).

tff(fact_3896_real__root__Suc__0,axiom,
    ! [X: real] : hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X) = X ).

tff(fact_3897_real__root__eq__iff,axiom,
    ! [X_1: real,Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2) )
      <=> ( X_1 = Y_2 ) ) ) ).

tff(fact_3898_real__root__commute,axiom,
    ! [X: real,N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_n546711566l_real(root,M),X)) ) ) ) ).

tff(fact_3899_odd__real__root__unique,axiom,
    ! [Y: real,X: real,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) = X )
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N),X) = Y ) ) ) ).

tff(fact_3900_odd__real__root__pow,axiom,
    ! [X: real,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X ) ) ).

tff(fact_3901_odd__real__root__power__cancel,axiom,
    ! [X: real,N: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = X ) ) ).

tff(fact_3902_real__root__le__iff,axiom,
    ! [X_1: real,Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ).

tff(fact_3903_real__root__le__mono,axiom,
    ! [X: real,Y: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))) ) ) ).

tff(fact_3904_real__root__eq__0__iff,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = zero_zero_real )
      <=> ( X_1 = zero_zero_real ) ) ) ).

tff(fact_3905_real__root__less__iff,axiom,
    ! [X_1: real,Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) ) ) ).

tff(fact_3906_real__root__less__mono,axiom,
    ! [X: real,Y: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))) ) ) ).

tff(fact_3907_real__root__mult,axiom,
    ! [X: real,Y: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) ) ) ).

tff(fact_3908_real__root__mult__exp,axiom,
    ! [X: real,N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ) ).

tff(fact_3909_real__root__power,axiom,
    ! [X: real,K_2: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),K_2) ) ) ).

tff(fact_3910_real__root__one,axiom,
    ! [N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),one_one_real) = one_one_real ) ) ).

tff(fact_3911_real__root__eq__1__iff,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = one_one_real )
      <=> ( X_1 = one_one_real ) ) ) ).

tff(fact_3912_real__root__divide,axiom,
    ! [X: real,Y: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) ) ) ).

tff(fact_3913_real__root__minus,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ).

tff(fact_3914_real__root__abs,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(abs_abs_real,X)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ).

tff(fact_3915_real__root__inverse,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ).

tff(fact_3916_real__root__ge__0__iff,axiom,
    ! [Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) ) ) ).

tff(fact_3917_real__root__le__0__iff,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) ) ) ).

tff(fact_3918_real__root__ge__zero,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ).

tff(fact_3919_real__root__gt__0__iff,axiom,
    ! [Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) ) ) ).

tff(fact_3920_real__root__lt__0__iff,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),zero_zero_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) ) ) ).

tff(fact_3921_real__root__gt__zero,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ).

tff(fact_3922_real__root__ge__1__iff,axiom,
    ! [Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) ) ) ).

tff(fact_3923_real__root__le__1__iff,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),one_one_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) ) ) ).

tff(fact_3924_real__root__decreasing,axiom,
    ! [X: real,N_5: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ) ).

tff(fact_3925_real__root__gt__1__iff,axiom,
    ! [Y_2: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) ) ) ).

tff(fact_3926_real__root__lt__1__iff,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),one_one_real))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) ) ) ).

tff(fact_3927_real__root__strict__decreasing,axiom,
    ! [X: real,N_5: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ) ).

tff(fact_3928_real__root__pos,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(suc,N))) = X ) ) ).

tff(fact_3929_real__root__pos__pos,axiom,
    ! [X_4: real,N_6: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_4))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_6),X_4))) ) ) ).

tff(fact_3930_real__root__less__mono__lemma,axiom,
    ! [Y: real,X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))) ) ) ) ).

tff(fact_3931_real__root__mult__lemma,axiom,
    ! [Y: real,X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) ) ) ) ) ).

tff(fact_3932_real__root__pos__mult__exp,axiom,
    ! [X: real,N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ) ) ).

tff(fact_3933_real__root__pos2,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = X ) ) ) ).

tff(fact_3934_real__root__pow__pos2,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X ) ) ) ).

tff(fact_3935_real__root__pos__unique,axiom,
    ! [X: real,Y: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) = X )
         => ( hAPP_real_real(hAPP_n546711566l_real(root,N),X) = Y ) ) ) ) ).

tff(fact_3936_real__root__pow__pos,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X ) ) ) ).

tff(fact_3937_real__root__increasing,axiom,
    ! [X: real,N_5: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X))) ) ) ) ) ).

tff(fact_3938_real__root__strict__increasing,axiom,
    ! [X: real,N_5: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X))) ) ) ) ) ).

tff(fact_3939_sqrt__def,axiom,
    sqrt = hAPP_n546711566l_real(root,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_3940_real__root__inverse__lemma,axiom,
    ! [X: real,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ) ).

tff(fact_3941_DERIV__odd__real__root,axiom,
    ! [X_1: real,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
     => ( ( X_1 != zero_zero_real )
       => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).

tff(fact_3942_DERIV__real__root,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).

tff(fact_3943_exp__total,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X_2: real] : hAPP_real_real(exp_real,X_2) = Y ) ).

tff(fact_3944_pdivmod__posDivAlg,axiom,
    ! [K_2: int,L_2: int] :
      ( ( ( L_2 = zero_zero_int )
       => ( pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(abs_abs_int,K_2)) ) )
      & ( ( L_2 != zero_zero_int )
       => ( pdivmod(K_2,L_2) = posDivAlg(hAPP_int_int(abs_abs_int,K_2),hAPP_int_int(abs_abs_int,L_2)) ) ) ) ).

tff(fact_3945_MVT2,axiom,
    ! [F: fun_real_real,F_1: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => hBOOL(hAPP_real_bool(deriv_real(F,X_2),hAPP_real_real(F_1,X_2))) )
       => ? [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Z))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),B))
            & ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_real_real(F_1,Z)) ) ) ) ) ).

tff(fact_3946_pdivmod__def,axiom,
    ! [K_2: int,L_2: int] : pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))) ).

tff(fact_3947_polar__ex2,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real))
     => ? [R_1: real,A_2: real] :
          ( ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) )
          & ( Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ) ).

tff(fact_3948_polar__ex1,axiom,
    ! [X: real,Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ? [R_1: real,A_2: real] :
          ( ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) )
          & ( Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ) ).

tff(fact_3949_z3mod__def,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),L_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) ) ) ) ).

tff(fact_3950_z3div__def,axiom,
    ! [K_2: int,L_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2))
       => ( hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),hAPP_int_int(uminus_uminus_int,L_2))) ) ) ) ).

tff(fact_3951_DERIV__isconst__all,axiom,
    ! [X_1: real,Y_2: real,F: fun_real_real] :
      ( ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real))
     => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_2) ) ) ).

tff(fact_3952_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_2))) ).

tff(fact_3953_le__Suc__ex,axiom,
    ! [K_2: nat,L_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
     => ? [N_2: nat] : L_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2) ) ).

tff(fact_3954_complex__norm,axiom,
    ! [X: real,Y: real] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_3955_exists__least__lemma,axiom,
    ! [P_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
     => ( ? [X1: nat] : hBOOL(hAPP_nat_bool(P_1,X1))
       => ? [N_2: nat] :
            ( ~ hBOOL(hAPP_nat_bool(P_1,N_2))
            & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) ) ) ) ).

tff(fact_3956_complex_Oinject,axiom,
    ! [Real1_2: real,Real2_2: real,Real1_1: real,Real2_1: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1_2),Real2_2) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1_1),Real2_1) )
    <=> ( ( Real1_2 = Real1_1 )
        & ( Real2_2 = Real2_1 ) ) ) ).

tff(fact_3957_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

tff(fact_3958_complex__zero__def,axiom,
    zero_zero_complex = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),zero_zero_real) ).

tff(fact_3959_Complex__eq__number__of,axiom,
    ! [A: real,B: real,W_1: int] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,A),B) = hAPP_int_complex(number528085621omplex,W_1) )
    <=> ( ( A = hAPP_int_real(number267125858f_real,W_1) )
        & ( B = zero_zero_real ) ) ) ).

tff(fact_3960_complex__one__def,axiom,
    one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex_1,one_one_real),zero_zero_real) ).

tff(fact_3961_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

tff(fact_3962_complex__add,axiom,
    ! [A_3: real,B_2: real,C_1: real,D_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)) ).

tff(fact_3963_complex__minus,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(uminus473333897omplex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,A_3)),hAPP_real_real(uminus_uminus_real,B_2)) ).

tff(fact_3964_complex__diff,axiom,
    ! [A_3: real,B_2: real,C_1: real,D_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),D_3)) ).

tff(fact_3965_complex__mult,axiom,
    ! [A_3: real,B_2: real,C_1: real,D_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),D_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),D_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),C_1))) ).

tff(fact_3966_cmod__unit__one,axiom,
    ! [A_3: real] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3))) = one_one_real ).

tff(fact_3967_complex__inverse,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(invers1449016382omplex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,B_2)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_3968_complex_Osize_I1_J,axiom,
    ! [Real1: real,Real2: real] : complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1),Real2)) = zero_zero_nat ).

tff(fact_3969_cos__arg__i__mult__zero__pos,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
     => ( hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),Y))) = zero_zero_real ) ) ).

tff(fact_3970_cos__arg__i__mult__zero__neg,axiom,
    ! [Y: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real))
     => ( hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),Y))) = zero_zero_real ) ) ).

tff(fact_3971_complex_Osize_I2_J,axiom,
    ! [Real1: real,Real2: real] : hAPP_complex_nat(size_size_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,Real1),Real2)) = zero_zero_nat ).

tff(fact_3972_complex__divide__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),hAPP_complex_complex(invers1449016382omplex,Y)) ).

tff(fact_3973_cos__arg__i__mult__zero,axiom,
    ! [Y: real] :
      ( ( Y != zero_zero_real )
     => ( hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),Y))) = zero_zero_real ) ) ).

tff(fact_3974_power2__i,axiom,
    hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,ii),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_complex(number528085621omplex,min) ).

tff(fact_3975_complex__i__not__zero,axiom,
    ii != zero_zero_complex ).

tff(fact_3976_complex__i__not__one,axiom,
    ii != one_one_complex ).

tff(fact_3977_complex__i__not__number__of,axiom,
    ! [W: int] : ii != hAPP_int_complex(number528085621omplex,W) ).

tff(fact_3978_complex__i__mult__minus,axiom,
    ! [X: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X)) = hAPP_complex_complex(uminus473333897omplex,X) ).

tff(fact_3979_inverse__i,axiom,
    hAPP_complex_complex(invers1449016382omplex,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).

tff(fact_3980_i__def,axiom,
    ii = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),one_one_real) ).

tff(fact_3981_Complex__eq__i,axiom,
    ! [X_1: real,Y_2: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,X_1),Y_2) = ii )
    <=> ( ( X_1 = zero_zero_real )
        & ( Y_2 = one_one_real ) ) ) ).

tff(fact_3982_Complex__mult__i,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,B_2)),A_3) ).

tff(fact_3983_i__mult__Complex,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,B_2)),A_3) ).

tff(fact_3984_i__squared,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_int_complex(number528085621omplex,min) ).

tff(fact_3985_i__mult__eq2,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).

tff(fact_3986_complex__inverse__complex__split,axiom,
    ! [X: real,Y: real] : hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).

tff(fact_3987_complex__eq__cancel__iff2,axiom,
    ! [X_1: real,Y_2: real,Xa_1: real] :
      ( ( hAPP_real_complex(hAPP_r265291036omplex(complex_1,X_1),Y_2) = of_real_complex(Xa_1) )
    <=> ( ( X_1 = Xa_1 )
        & ( Y_2 = zero_zero_real ) ) ) ).

tff(fact_3988_complex__of__real__def,axiom,
    ! [R_3: real] : of_real_complex(R_3) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,R_3),zero_zero_real) ).

tff(fact_3989_sgn__eq,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(sgn_sgn_complex,Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Z_1),of_real_complex(hAPP_complex_real(norm_norm_complex,Z_1))) ).

tff(fact_3990_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),R_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),R_3)) ).

tff(fact_3991_complex__of__real__mult__Complex,axiom,
    ! [R_3: real,X: real,Y: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),Y)) ).

tff(fact_3992_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),R_3)),Y) ).

tff(fact_3993_complex__of__real__add__Complex,axiom,
    ! [R_3: real,X: real,Y: real] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),X)),Y) ).

tff(fact_3994_i__mult__eq,axiom,
    hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = of_real_complex(hAPP_int_real(number267125858f_real,min)) ).

tff(fact_3995_complex__of__real__i,axiom,
    ! [R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),R_3) ).

tff(fact_3996_i__complex__of__real,axiom,
    ! [R_3: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,zero_zero_real),R_3) ).

tff(fact_3997_complex__of__real__minus__one,axiom,
    of_real_complex(hAPP_real_real(uminus_uminus_real,one_one_real)) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).

tff(fact_3998_cmod__complex__polar,axiom,
    ! [R_3: real,A_3: real] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)))) = hAPP_real_real(abs_abs_real,R_3) ).

tff(fact_3999_expi__two__pi__i,axiom,
    expi(hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),of_real_complex(pi))),ii)) = one_one_complex ).

tff(fact_4000_expi__add,axiom,
    ! [A_3: complex,B_2: complex] : expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_3)),expi(B_2)) ).

tff(fact_4001_expi__zero,axiom,
    expi(zero_zero_complex) = one_one_complex ).

tff(fact_4002_complex__split__polar,axiom,
    ! [Z_1: complex] :
    ? [R_1: real,A_2: real] : Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2))) ).

tff(fact_4003_DERIV__pos__imp__increasing,axiom,
    ! [F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,A)),hAPP_real_real(F,B))) ) ) ).

tff(fact_4004_DERIV__neg__imp__decreasing,axiom,
    ! [F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))) ) ) ).

tff(fact_4005_int__induct,axiom,
    ! [I_1: int,P_1: fun_int_bool,K_1: int] :
      ( hBOOL(hAPP_int_bool(P_1,K_1))
     => ( ! [I_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2))
           => ( hBOOL(hAPP_int_bool(P_1,I_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) ) )
       => ( ! [I_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1))
             => ( hBOOL(hAPP_int_bool(P_1,I_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(P_1,I_1)) ) ) ) ).

tff(fact_4006_minusinfinity,axiom,
    ! [P_1: fun_int_bool,P1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int,K: int] :
            ( hBOOL(hAPP_int_bool(P1,X_2))
          <=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z: int] :
            ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
              <=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
         => ( ? [X1: int] : hBOOL(hAPP_int_bool(P1,X1))
           => ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).

tff(fact_4007_plusinfinity,axiom,
    ! [P_1: fun_int_bool,P_2: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int,K: int] :
            ( hBOOL(hAPP_int_bool(P_2,X_2))
          <=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [Z: int] :
            ! [X_2: int] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
              <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
         => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_2,X1))
           => ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ).

tff(fact_4008_coprime__sos,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_4009_coprime__lmul2,axiom,
    ! [D_3: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)) ) ).

tff(fact_4010_coprime__rmul2,axiom,
    ! [D_3: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) ) ).

tff(fact_4011_coprime__mul,axiom,
    ! [B_2: nat,D_3: nat,A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) ) ) ).

tff(fact_4012_coprime__mul__eq,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),A))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),B)) ) ) ).

tff(fact_4013_coprime__refl,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,N_1),N_1))
    <=> ( N_1 = one_one_nat ) ) ).

tff(fact_4014_coprime__1,axiom,
    ! [A_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),one_one_nat)) ).

tff(fact_4015_coprime__1_H,axiom,
    ! [A_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_3)) ).

tff(fact_4016_coprime__commute,axiom,
    ! [A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B),A)) ) ).

tff(fact_4017_coprime__divisors,axiom,
    ! [E_1: nat,B_2: nat,D_3: nat,A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E_1),B_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),E_1)) ) ) ) ).

tff(fact_4018_coprime__exp__imp,axiom,
    ! [N: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))) ) ).

tff(fact_4019_coprime__exp,axiom,
    ! [N: nat,D_3: nat,A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N))) ) ).

tff(fact_4020_floor__real__of__nat,axiom,
    ! [N: nat] : archim1246769320r_real(hAPP_nat_real(real_nat,N)) = hAPP_nat_int(semiri1621563631at_int,N) ).

tff(fact_4021_natfloor__def,axiom,
    ! [X: real] : natfloor(X) = hAPP_int_nat(nat_1,archim1246769320r_real(X)) ).

tff(fact_4022_coprime__Suc0_H,axiom,
    ! [A_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(suc,zero_zero_nat)),A_3)) ).

tff(fact_4023_coprime__Suc0,axiom,
    ! [A_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_4024_coprime__0,axiom,
    ! [D: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),zero_zero_nat))
    <=> ( D = one_one_nat ) ) ).

tff(fact_4025_coprime__0_H,axiom,
    ! [D: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D))
    <=> ( D = one_one_nat ) ) ).

tff(fact_4026_divides__mul,axiom,
    ! [N: nat,M: nat,R_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_3))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,M),N))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),R_3)) ) ) ) ).

tff(fact_4027_coprime__divprod,axiom,
    ! [D_3: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2)) ) ) ).

tff(fact_4028_coprime,axiom,
    ! [A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
    <=> ! [D_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) )
        <=> ( D_2 = one_one_nat ) ) ) ).

tff(fact_4029_coprime__plus1,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N)) ).

tff(fact_4030_coprime__exp2,axiom,
    ! [A: nat,N_1: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),hAPP_nat_nat(suc,N_1))))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)) ) ).

tff(fact_4031_coprime__minus1,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

tff(fact_4032_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))))
       => ( hAPP_int_nat(nat_1,archim1246769320r_real(X)) = N ) ) ) ).

tff(fact_4033_floor__minus__real__of__nat,axiom,
    ! [N: nat] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N)) ).

tff(fact_4034_le__mult__floor,axiom,
    ! [B_2: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,archim1246769320r_real(A_3)),archim1246769320r_real(B_2))),archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)))) ) ) ).

tff(fact_4035_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))))
       => ( hAPP_int_nat(nat_1,archim1246769320r_real(X)) = N ) ) ) ).

tff(fact_4036_coprime__bezout,axiom,
    ! [A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
    <=> ? [X_2: nat,Y_1: nat] :
          ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),Y_1)) = one_one_nat )
          | ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),Y_1)) = one_one_nat ) ) ) ).

tff(fact_4037_coprime__bezout__strong,axiom,
    ! [A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
     => ( ( B_2 != one_one_nat )
       => ? [X_2: nat,Y_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y_1)),one_one_nat) ) ) ).

tff(fact_4038_nat__number__of__diff__1,axiom,
    ! [V_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2)) ) ) ) ).

tff(fact_4039_succ__pred,axiom,
    ! [X: int] : hAPP_int_int(succ,hAPP_int_int(pred,X)) = X ).

tff(fact_4040_pred__Bit1,axiom,
    ! [K_2: int] : hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2) ).

tff(fact_4041_pred__Bit0,axiom,
    ! [K_2: int] : hAPP_int_int(pred,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,K_2)) ).

tff(fact_4042_le__iff__pred__less,axiom,
    ! [K_1: int,L: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K_1)),L)) ) ).

tff(fact_4043_pred__Pls,axiom,
    hAPP_int_int(pred,pls) = min ).

tff(fact_4044_pred__Min,axiom,
    hAPP_int_int(pred,min) = hAPP_int_int(bit0,min) ).

tff(fact_4045_pred__def,axiom,
    ! [K_2: int] : hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),one_one_int) ).

tff(fact_4046_minus__Bit1,axiom,
    ! [K_2: int] : hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,hAPP_int_int(uminus_uminus_int,K_2))) ).

tff(fact_4047_add__Min__right,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),min) = hAPP_int_int(pred,K_2) ).

tff(fact_4048_add__Min,axiom,
    ! [K_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,min),K_2) = hAPP_int_int(pred,K_2) ).

tff(fact_4049_diff__bin__simps_I8_J,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(pred,K_2)),L_2)) ).

tff(fact_4050_neg__number__of__pred__iff__0,axiom,
    ! [V: int] :
      ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
    <=> ( hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat ) ) ).

tff(fact_4051_Suc__diff__number__of,axiom,
    ! [M: nat,V_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2))) ) ) ).

tff(fact_4052_complex__expi__Ex,axiom,
    ! [Z_1: complex] :
    ? [A_2: complex,R_1: real] : Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_2)) ).

tff(fact_4053_ResSet__finite,axiom,
    ! [X_3: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_3)) ) ) ).

tff(fact_4054_ResSet__def,axiom,
    ! [Ma: int,X_3: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3))
    <=> ! [Y1: int,Y2: int] :
          ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_3))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_3))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y1),Y2),Ma)) )
         => ( Y1 = Y2 ) ) ) ).

tff(fact_4055_ResSet__SRStar__prop,axiom,
    ! [P_5: int] : hBOOL(hAPP_f448129468l_bool(resSet(P_5),hAPP_i1948725293t_bool(sRStar,P_5))) ).

tff(fact_4056_nat_Osize_I2_J,axiom,
    ! [Nat: nat] : hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4057_nat_Osize_I4_J,axiom,
    ! [Nat: nat] : hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4058_code__numeral_Osize_I1_J,axiom,
    code_c271388182l_size(zero_z126310315umeral) = zero_zero_nat ).

tff(fact_4059_nat__size,axiom,
    ! [N: nat] : hAPP_nat_nat(size_size_nat,N) = N ).

tff(fact_4060_nat_Osize_I3_J,axiom,
    hAPP_nat_nat(size_size_nat,zero_zero_nat) = zero_zero_nat ).

tff(fact_4061_nat_Osize_I1_J,axiom,
    hAPP_nat_nat(nat_size,zero_zero_nat) = zero_zero_nat ).

tff(fact_4062_code__numeral_Osize_I2_J,axiom,
    ! [Code_numeral_1: code_code_numeral] : code_c271388182l_size(code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,code_c271388182l_size(Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4063_complex__mod__mult__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(norm_norm_complex,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).

tff(fact_4064_complex__cnj__inverse,axiom,
    ! [X: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X)) ).

tff(fact_4065_complex__cnj__zero__iff,axiom,
    ! [Z_2: complex] :
      ( ( hAPP_complex_complex(cnj,Z_2) = zero_zero_complex )
    <=> ( Z_2 = zero_zero_complex ) ) ).

tff(fact_4066_cnj_Ozero,axiom,
    hAPP_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

tff(fact_4067_complex__cnj__number__of,axiom,
    ! [W: int] : hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W)) = hAPP_int_complex(number528085621omplex,W) ).

tff(fact_4068_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4069_complex__cnj__one,axiom,
    hAPP_complex_complex(cnj,one_one_complex) = one_one_complex ).

tff(fact_4070_cnj_Ominus,axiom,
    ! [X: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X)) ).

tff(fact_4071_cnj_Oadd,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4072_complex__cnj__cancel__iff,axiom,
    ! [X_1: complex,Y_2: complex] :
      ( ( hAPP_complex_complex(cnj,X_1) = hAPP_complex_complex(cnj,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_4073_complex__cnj__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1)) = Z_1 ).

tff(fact_4074_code__numeral_Oinject,axiom,
    ! [Code_numeral_5: code_code_numeral,Code_numeral_4: code_code_numeral] :
      ( ( code_S1047413653umeral(Code_numeral_5) = code_S1047413653umeral(Code_numeral_4) )
    <=> ( Code_numeral_5 = Code_numeral_4 ) ) ).

tff(fact_4075_complex__cnj__of__nat,axiom,
    ! [N: nat] : hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_complex(semiri2020571505omplex,N) ).

tff(fact_4076_complex__mod__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_real(norm_norm_complex,Z_1) ).

tff(fact_4077_cnj_Odiff,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4078_complex__cnj__power,axiom,
    ! [X: complex,N: nat] : hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X),N)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X)),N) ).

tff(fact_4079_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).

tff(fact_4080_complex__cnj__complex__of__real,axiom,
    ! [X: real] : hAPP_complex_complex(cnj,of_real_complex(X)) = of_real_complex(X) ).

tff(fact_4081_code__numeral_Osimps_I3_J,axiom,
    ! [Code_numeral_3: code_code_numeral] : code_S1047413653umeral(Code_numeral_3) != zero_z126310315umeral ).

tff(fact_4082_code__numeral_Osimps_I2_J,axiom,
    ! [Code_numeral_2: code_code_numeral] : zero_z126310315umeral != code_S1047413653umeral(Code_numeral_2) ).

tff(fact_4083_complex__cnj,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(cnj,hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),hAPP_real_real(uminus_uminus_real,B_2)) ).

tff(fact_4084_complex__cnj__i,axiom,
    hAPP_complex_complex(cnj,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).

tff(fact_4085_Suc__code__numeral__minus__one,axiom,
    ! [N: code_code_numeral] : hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N)),one_on1645066479umeral) = N ).

tff(fact_4086_cnj_Opos__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4087_cnj_Ononneg__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4088_cnj_Obounded,axiom,
    ? [K_4: real] :
    ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ).

tff(fact_4089_code__numeral_Osize_I4_J,axiom,
    ! [Code_numeral_1: code_code_numeral] : hAPP_C2129356693al_nat(size_s945831648umeral,code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_C2129356693al_nat(size_s945831648umeral,Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).

tff(fact_4090_complex__diff__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(im,Z_1)))),ii) ).

tff(fact_4091_code__numeral_Oexhaust,axiom,
    ! [Y: code_code_numeral] :
      ( ( Y != zero_z126310315umeral )
     => ~ ! [Code_numeral: code_code_numeral] : Y != code_S1047413653umeral(Code_numeral) ) ).

tff(fact_4092_Im,axiom,
    ! [X: real,Y: real] : hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = Y ).

tff(fact_4093_complex__Im__cnj,axiom,
    ! [X: complex] : hAPP_complex_real(im,hAPP_complex_complex(cnj,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) ).

tff(fact_4094_Im__complex__of__real,axiom,
    ! [Z_1: real] : hAPP_complex_real(im,of_real_complex(Z_1)) = zero_zero_real ).

tff(fact_4095_complex__Im__i,axiom,
    hAPP_complex_real(im,ii) = one_one_real ).

tff(fact_4096_Im_Ozero,axiom,
    hAPP_complex_real(im,zero_zero_complex) = zero_zero_real ).

tff(fact_4097_complex__Im__one,axiom,
    hAPP_complex_real(im,one_one_complex) = zero_zero_real ).

tff(fact_4098_Im_Oadd,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) ).

tff(fact_4099_complex__Im__number__of,axiom,
    ! [V_2: int] : hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_2)) = zero_zero_real ).

tff(fact_4100_Im_Ominus,axiom,
    ! [X: complex] : hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) ).

tff(fact_4101_Im_Odiff,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) ).

tff(fact_4102_complex__Im__of__nat,axiom,
    ! [N: nat] : hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N)) = zero_zero_real ).

tff(fact_4103_abs__Im__le__cmod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X))),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_4104_complex__In__mult__cnj__zero,axiom,
    ! [Z_1: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = zero_zero_real ).

tff(fact_4105_Im__sgn,axiom,
    ! [Z_1: complex] : hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(im,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) ).

tff(fact_4106_code__numeral_Osize_I3_J,axiom,
    hAPP_C2129356693al_nat(size_s945831648umeral,zero_z126310315umeral) = zero_zero_nat ).

tff(fact_4107_Im_Opos__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4108_Im_Ononneg__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4109_Im_Obounded,axiom,
    ? [K_4: real] :
    ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ).

tff(fact_4110_complex__inverse__def,axiom,
    ! [X: complex] : hAPP_complex_complex(invers1449016382omplex,X) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_4111_complex__mult__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4112_complex__Im__inverse,axiom,
    ! [X: complex] : hAPP_complex_real(im,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4113_Re,axiom,
    ! [X: real,Y: real] : hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex_1,X),Y)) = X ).

tff(fact_4114_Re__complex__of__real,axiom,
    ! [Z_1: real] : hAPP_complex_real(re,of_real_complex(Z_1)) = Z_1 ).

tff(fact_4115_complex__Re__cnj,axiom,
    ! [X: complex] : hAPP_complex_real(re,hAPP_complex_complex(cnj,X)) = hAPP_complex_real(re,X) ).

tff(fact_4116_complex__eq__iff,axiom,
    ! [X_1: complex,Y_2: complex] :
      ( ( X_1 = Y_2 )
    <=> ( ( hAPP_complex_real(re,X_1) = hAPP_complex_real(re,Y_2) )
        & ( hAPP_complex_real(im,X_1) = hAPP_complex_real(im,Y_2) ) ) ) ).

tff(fact_4117_complex__eqI,axiom,
    ! [X: complex,Y: complex] :
      ( ( hAPP_complex_real(re,X) = hAPP_complex_real(re,Y) )
     => ( ( hAPP_complex_real(im,X) = hAPP_complex_real(im,Y) )
       => ( X = Y ) ) ) ).

tff(fact_4118_complex__Re__le__cmod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(re,X)),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_4119_complex__Re__i,axiom,
    hAPP_complex_real(re,ii) = zero_zero_real ).

tff(fact_4120_complex__surj,axiom,
    ! [Z_1: complex] : hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_complex_real(re,Z_1)),hAPP_complex_real(im,Z_1)) = Z_1 ).

tff(fact_4121_Re_Ozero,axiom,
    hAPP_complex_real(re,zero_zero_complex) = zero_zero_real ).

tff(fact_4122_Re_Oadd,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) ).

tff(fact_4123_complex__Re__one,axiom,
    hAPP_complex_real(re,one_one_complex) = one_one_real ).

tff(fact_4124_Re_Ominus,axiom,
    ! [X: complex] : hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X)) ).

tff(fact_4125_complex__Re__number__of,axiom,
    ! [V_2: int] : hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_2)) = hAPP_int_real(number267125858f_real,V_2) ).

tff(fact_4126_Re_Odiff,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) ).

tff(fact_4127_complex__Re__of__nat,axiom,
    ! [N: nat] : hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_real(semiri132038758t_real,N) ).

tff(fact_4128_abs__Re__le__cmod,axiom,
    ! [X: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(re,X))),hAPP_complex_real(norm_norm_complex,X))) ).

tff(fact_4129_Re__sgn,axiom,
    ! [Z_1: complex] : hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) ).

tff(fact_4130_complex__Im__mult,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y))) ).

tff(fact_4131_complex__Re__mult,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) ).

tff(fact_4132_cnj__def,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(cnj,Z_1) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_complex_real(re,Z_1)),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,Z_1))) ).

tff(fact_4133_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) ).

tff(fact_4134_complex__add__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) ).

tff(fact_4135_complex__minus__def,axiom,
    ! [X: complex] : hAPP_complex_complex(uminus473333897omplex,X) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X))),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))) ).

tff(fact_4136_complex__mult__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y)))) ).

tff(fact_4137_complex__add__cnj,axiom,
    ! [Z_1: complex] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(re,Z_1))) ).

tff(fact_4138_cmod__def,axiom,
    ! [Z_1: complex] : hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4139_complex__Re__inverse,axiom,
    ! [X: complex] : hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4140_Re_Opos__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4141_Re_Ononneg__bounded,axiom,
    ? [K_4: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4))
      & ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ) ).

tff(fact_4142_Re_Obounded,axiom,
    ? [K_4: real] :
    ! [X_2: complex] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4))) ).

tff(fact_4143_expi__def,axiom,
    ! [Z_1: complex] : expi(Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(exp_real,hAPP_complex_real(re,Z_1)))),cis(hAPP_complex_real(im,Z_1))) ).

tff(fact_4144_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A_3: real] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) = hAPP_complex_real(im,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) ).

tff(fact_4145_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A_3: real] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) = hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) ).

tff(fact_4146_Re__cis,axiom,
    ! [A_3: real] : hAPP_complex_real(re,cis(A_3)) = hAPP_real_real(cos,A_3) ).

tff(fact_4147_Im__cis,axiom,
    ! [A_3: real] : hAPP_complex_real(im,cis(A_3)) = hAPP_real_real(sin,A_3) ).

tff(fact_4148_cis__mult,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(B_2)) = cis(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) ).

tff(fact_4149_cis__zero,axiom,
    cis(zero_zero_real) = one_one_complex ).

tff(fact_4150_cis__inverse,axiom,
    ! [A_3: real] : hAPP_complex_complex(invers1449016382omplex,cis(A_3)) = cis(hAPP_real_real(uminus_uminus_real,A_3)) ).

tff(fact_4151_cis__divide,axiom,
    ! [A_3: real,B_2: real] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,cis(A_3)),cis(B_2)) = cis(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).

tff(fact_4152_cis__def,axiom,
    ! [A_3: real] : cis(A_3) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)) ).

tff(fact_4153_DeMoivre,axiom,
    ! [A_3: real,N: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) ).

tff(fact_4154_cis__real__of__nat__Suc__mult,axiom,
    ! [N: nat,A_3: real] : cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))),A_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) ).

tff(fact_4155_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

tff(fact_4156_termination__basic__simps_I4_J,axiom,
    ! [Y: nat,X: nat,Z_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_4157_termination__basic__simps_I1_J,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_4158_termination__basic__simps_I2_J,axiom,
    ! [Y: nat,X: nat,Z_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_4159_termination__basic__simps_I5_J,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).

tff(fact_4160_termination__basic__simps_I3_J,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).

tff(fact_4161_prime__dvd__power__two,axiom,
    ! [M: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M)) ) ) ).

tff(fact_4162_divmod__int__def,axiom,
    ! [B_2: int,A_3: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
           => ( divmod_int(A_3,B_2) = posDivAlg(A_3,B_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
           => ( ( ( A_3 = zero_zero_int )
               => ( divmod_int(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ) )
              & ( ( A_3 != zero_zero_int )
               => ( divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) ) ) ) ) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3))
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
           => ( divmod_int(A_3,B_2) = negDivAlg(A_3,B_2) ) )
          & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
           => ( divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) ) ) ) ) ) ).

tff(fact_4163_prime__0,axiom,
    ~ hBOOL(hAPP_nat_bool(prime,zero_zero_nat)) ).

tff(fact_4164_prime__1,axiom,
    ~ hBOOL(hAPP_nat_bool(prime,one_one_nat)) ).

tff(fact_4165_primes__eq,axiom,
    ! [Q_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(prime,Q_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_2))
         => ( P_3 = Q_2 ) ) ) ) ).

tff(fact_4166_distinct__prime__coprime,axiom,
    ! [Q_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(prime,Q_2))
       => ( ( P_3 != Q_2 )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_2)) ) ) ) ).

tff(fact_4167_prime__Suc0,axiom,
    ~ hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(suc,zero_zero_nat))) ).

tff(fact_4168_prime__g__zero,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),P_3)) ) ).

tff(fact_4169_prime__divprod__eq,axiom,
    ! [A: nat,B: nat,P_5: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_5))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)))
      <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),A))
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),B)) ) ) ) ).

tff(fact_4170_prime__divprod,axiom,
    ! [A_3: nat,B_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) ) ) ) ).

tff(fact_4171_prime__dvd__mult,axiom,
    ! [M: nat,N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N)) ) ) ) ).

tff(fact_4172_prime__product,axiom,
    ! [P_3: nat,Q_2: nat] :
      ( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_2)))
     => ( ( P_3 = one_one_nat )
        | ( Q_2 = one_one_nat ) ) ) ).

tff(fact_4173_prime__divexp__n,axiom,
    ! [X: nat,N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) ) ) ).

tff(fact_4174_prime__divexp,axiom,
    ! [X: nat,N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X)) ) ) ).

tff(fact_4175_prime__dvd__power,axiom,
    ! [A_3: nat,N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) ) ) ).

tff(fact_4176_prime__exp,axiom,
    ! [P_5: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(prime,P_5))
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_4177_coprime__prime,axiom,
    ! [P_3: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
     => ~ ( hBOOL(hAPP_nat_bool(prime,P_3))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) ) ) ).

tff(fact_4178_prime__coprime__strong,axiom,
    ! [N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) ) ) ).

tff(fact_4179_coprime__prime__eq,axiom,
    ! [A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B))
    <=> ! [P_4: nat] :
          ~ ( hBOOL(hAPP_nat_bool(prime,P_4))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),A))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),B)) ) ) ).

tff(fact_4180_prime__impl__zprime__int,axiom,
    ! [A_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,A_3))
     => hBOOL(hAPP_int_bool(zprime,hAPP_nat_int(semiri1621563631at_int,A_3))) ) ).

tff(fact_4181_prime__g__one,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),P_3)) ) ).

tff(fact_4182_prime__nd__one,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(suc,zero_zero_nat))) ) ).

tff(fact_4183_prime__factor__lt,axiom,
    ! [M: nat,N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( N != zero_zero_nat )
       => ( ( N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),M) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ) ).

tff(fact_4184_prime__def,axiom,
    ! [P_5: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_5))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),P_5))
        & ! [M_1: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),P_5))
           => ( ( M_1 = one_one_nat )
              | ( M_1 = P_5 ) ) ) ) ) ).

tff(fact_4185_divides__primepow,axiom,
    ! [D: nat,K_1: nat,P_5: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_5))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),K_1)))
      <=> ? [I_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1))
            & ( D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),I_2) ) ) ) ) ).

tff(fact_4186_prime__power__dvd__cancel__right,axiom,
    ! [N: nat,A_3: nat,B_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),A_3)) ) ) ) ).

tff(fact_4187_prime__coprime__lt,axiom,
    ! [X: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),P_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),P_3)) ) ) ) ).

tff(fact_4188_prime__coprime,axiom,
    ! [N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( N = one_one_nat )
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) ) ) ).

tff(fact_4189_negateSnd__eq,axiom,
    ! [Q_2: int,R_3: int] : hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),hAPP_int_int(uminus_uminus_int,R_3)) ).

tff(fact_4190_divmod__int__rel__neg,axiom,
    ! [A_3: int,B_2: int,Qr_1: product_prod_int_int] :
      ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)),Qr_1))
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_P1975530577nt_int(negateSnd,Qr_1))) ) ).

tff(fact_4191_two__is__prime,axiom,
    hBOOL(hAPP_nat_bool(prime,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_4192_divmod__int__correct,axiom,
    ! [A_3: int,B_2: int] :
      ( ( B_2 != zero_zero_int )
     => hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),divmod_int(A_3,B_2))) ) ).

tff(fact_4193_prime__dvd__square,axiom,
    ! [M: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M)) ) ) ).

tff(fact_4194_prime__divprod__pow,axiom,
    ! [N: nat,A_3: nat,B_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),A_3))
            | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),B_2)) ) ) ) ) ).

tff(fact_4195_prime__ge__2,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) ) ).

tff(fact_4196_prime__odd,axiom,
    ! [P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( P_3 = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) )
        | ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)) ) ) ).

tff(fact_4197_divmod__int__mod__div,axiom,
    ! [P_3: int,Q_2: int] : divmod_int(P_3,Q_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_2)),hAPP_int_int(div_mod_int(P_3),Q_2)) ).

tff(fact_4198_div__mod__code__numeral__def,axiom,
    ! [N: code_code_numeral,M: code_code_numeral] : code_d418564891umeral(N,M) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N),M)),hAPP_C498520661umeral(div_mo1740067990umeral(N),M)) ).

tff(fact_4199_xzgcd__linear,axiom,
    ! [M: int,R_3: int,S_1: int,T_1: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( ( xzgcd(M,N) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_1),T_1)) )
       => ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)) ) ) ) ).

tff(fact_4200_xzgcda_Osimps,axiom,
    ! [M: int,N: int,R_5: int,S_3: int,S_1: int,T_2: int,T_1: int,R_3: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
       => ( xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_5),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_2)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int))
       => ( xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) = xzgcda(M,N,R_3,hAPP_int_int(div_mod_int(R_5),R_3),S_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1)),T_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1))) ) ) ) ).

tff(fact_4201_xzgcd__def,axiom,
    ! [M: int,N: int] : xzgcd(M,N) = xzgcda(M,N,M,N,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).

tff(fact_4202_xzgcda__linear,axiom,
    ! [M: int,N: int,R_5: int,S_3: int,S_1: int,T_2: int,T_1: int,Rn: int,Sn_1: int,Tn_1: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
     => ( ( xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Rn),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn_1),Tn_1)) )
       => ( ( R_5 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N)) )
         => ( ( R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)) )
           => ( Rn = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Sn_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Tn_1),N)) ) ) ) ) ) ).

tff(fact_4203_coprime__prime__dvd__ex,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y))
     => ? [P_4: nat] :
          ( hBOOL(hAPP_nat_bool(prime,P_4))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Y)) ) ) ).

tff(fact_4204_prime__power__exp,axiom,
    ! [X: nat,K_2: nat,N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( N != zero_zero_nat )
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) )
         => ? [I_2: nat] : X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) ) ) ) ).

tff(fact_4205_prime__factor,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ? [P_4: nat] :
          ( hBOOL(hAPP_nat_bool(prime,P_4))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),N)) ) ) ).

tff(fact_4206_divmod__nat__step,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
       => ( divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

tff(fact_4207_bezout__prime,axiom,
    ! [A_3: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))
       => ? [X_2: nat,Y_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Y_1)),one_one_nat) ) ) ).

tff(fact_4208_not__prime__ex__mk,axiom,
    ! [N: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
        & ~ hBOOL(hAPP_nat_bool(prime,N)) )
     => ? [M_1: nat,K: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
          & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N))
          & ( N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K) ) ) ) ).

tff(fact_4209_divmod__nat__zero,axiom,
    ! [M: nat] : divmod_nat(M,zero_zero_nat) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) ).

tff(fact_4210_divmod__nat__base,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
     => ( divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) ) ) ).

tff(fact_4211_divmod__nat__div__mod,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N)) ).

tff(fact_4212_divmod__nat__rel__mult1__eq,axiom,
    ! [A_3: nat,B_2: nat,C_1: nat,Q_2: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
       => hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),Q_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1)))) ) ) ).

tff(fact_4213_divmod__nat__rel__mult2__eq,axiom,
    ! [C_1: nat,A_3: nat,B_2: nat,Q_2: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,B_2),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_2))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
         => hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Q_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)))) ) ) ) ).

tff(fact_4214_divmod__nat__rel__add1__eq,axiom,
    ! [B_2: nat,Bq: nat,Br: nat,A_3: nat,C_1: nat,Aq: nat,Ar: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aq),Ar)))
     => ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Bq),Br)))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1))
         => hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aq),Bq)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1)))) ) ) ) ).

tff(fact_4215_divmod__nat__rel__unique,axiom,
    ! [Qr_2: product_prod_nat_nat,M: nat,N: nat,Qr_1: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1))
     => ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_2))
       => ( Qr_1 = Qr_2 ) ) ) ).

tff(fact_4216_divmod__nat__eq,axiom,
    ! [M: nat,N: nat,Qr_1: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1))
     => ( divmod_nat(M,N) = Qr_1 ) ) ).

tff(fact_4217_divmod__nat__rel__divmod__nat,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),divmod_nat(M,N))) ).

tff(fact_4218_div__eq,axiom,
    ! [M: nat,N: nat,Q_2: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = Q_2 ) ) ).

tff(fact_4219_mod__eq,axiom,
    ! [M: nat,N: nat,Q_2: nat,R_3: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3)))
     => ( hAPP_nat_nat(div_mod_nat(M),N) = R_3 ) ) ).

tff(fact_4220_divmod__nat__rel,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N)))) ).

tff(fact_4221_divmod__nat__rel__ex,axiom,
    ! [M: nat,N: nat] :
      ~ ! [Q_4: nat,R_1: nat] : ~ hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_4),R_1))) ).

tff(fact_4222_euclid,axiom,
    ! [N: nat] :
    ? [P_4: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_4))
      & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),P_4)) ) ).

tff(fact_4223_div__pos__neg__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) ) ) ).

tff(fact_4224_div__int__def,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,divmod_int(A_3,B_2)) ).

tff(fact_4225_div__neg__pos,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,negDivAlg(A_3,B_2)) ) ) ) ).

tff(fact_4226_div__pos__pos,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(A_3,B_2)) ) ) ) ).

tff(fact_4227_div__pos__pos__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) ) ) ).

tff(fact_4228_div__pos__neg,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ) ).

tff(fact_4229_div__neg__neg,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))
       => ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ) ).

tff(fact_4230_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) ) ) ) ).

tff(fact_4231_bezw__non__0,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) ) ) ).

tff(fact_4232_mod__pos__neg__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int))
     => ( hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) ) ) ).

tff(fact_4233_mod__int__def,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,divmod_int(A_3,B_2)) ).

tff(fact_4234_mod__neg__pos,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_3,B_2)) ) ) ) ).

tff(fact_4235_mod__pos__pos,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(A_3,B_2)) ) ) ) ).

tff(fact_4236_mod__pos__pos__1__number__of,axiom,
    ! [W: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
     => ( hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) ) ) ).

tff(fact_4237_mod__pos__neg,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ) ).

tff(fact_4238_mod__neg__neg,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))
       => ( hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) ) ) ) ).

tff(fact_4239_div__nat__def,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_P1654798560at_nat(product_fst_nat_nat,divmod_nat(M,N)) ).

tff(fact_4240_mod__nat__def,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(div_mod_nat(M),N) = hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M,N)) ).

tff(fact_4241_divmod__nat__rel__def,axiom,
    ! [Ma: nat,N_1: nat,Qr: product_prod_nat_nat] :
      ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(Ma,N_1),Qr))
    <=> ( ( Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)),N_1)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)) )
        & ( ( N_1 = zero_zero_nat )
         => ( hAPP_P1654798560at_nat(product_fst_nat_nat,Qr) = zero_zero_nat ) )
        & ( ( N_1 != zero_zero_nat )
         => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
             => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),N_1)) ) )
            & ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
             => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),zero_zero_nat)) ) ) ) ) ) ) ).

tff(fact_4242_Bolzano__bisect__diff,axiom,
    ! [P_1: fun_Pr1127657839l_bool,N_1: nat,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)) ) ) ).

tff(fact_4243_Bolzano__bisect__le,axiom,
    ! [P_1: fun_Pr1127657839l_bool,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2)))) ) ).

tff(fact_4244_Bolzano__bisect_Osimps_I1_J,axiom,
    ! [P_1: fun_Pr1127657839l_bool,A: real,B: real] : hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),zero_zero_nat) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B) ).

tff(fact_4245_Bolzano__bisect__fst__le__Suc,axiom,
    ! [P_1: fun_Pr1127657839l_bool,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2))))) ) ).

tff(fact_4246_Bolzano__bisect__Suc__le__snd,axiom,
    ! [P_1: fun_Pr1127657839l_bool,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2)))) ) ).

tff(fact_4247_not__P__Bolzano__bisect,axiom,
    ! [N_1: nat,A: real,B: real,P_1: fun_Pr1127657839l_bool] :
      ( ! [A_2: real,B_4: real,C_3: real] :
          ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
         => ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3)))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
             => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3))
               => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) ) ) ) )
     => ( ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B)))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
         => ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))))) ) ) ) ).

tff(fact_4248_not__P__Bolzano__bisect_H,axiom,
    ! [A: real,B: real,P_1: fun_Pr1127657839l_bool] :
      ( ! [A_2: real,B_4: real,C_3: real] :
          ( ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
            & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3)))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) )
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) )
     => ( ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B)))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
         => ! [N_2: nat] : ~ hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))) ) ) ) ).

tff(fact_4249_one__code__int__code,axiom,
    one_on1684967323de_int = hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit1,pls)) ).

tff(fact_4250_Quickcheck__Narrowing_Oint__of__code,axiom,
    ! [K_2: quickcheck_code_int] :
      ( ( ( K_2 = zero_z891286103de_int )
       => ( quickcheck_int_of(K_2) = zero_zero_int ) )
      & ( ( K_2 != zero_z891286103de_int )
       => ( ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int )
           => ( quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) )
          & ( ( hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z891286103de_int )
           => ( quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) ) ) ) ) ) ).

tff(fact_4251_DeMoivre2,axiom,
    ! [R_3: real,A_3: real,N: nat] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_3,A_3)),N) = rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) ).

tff(fact_4252_less__eq__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,Ma: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,N_1),Ma))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(N_1)),quickcheck_int_of(Ma))) ) ).

tff(fact_4253_less__code__int__def,axiom,
    ! [N_1: quickcheck_code_int,Ma: quickcheck_code_int] :
      ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,N_1),Ma))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(N_1)),quickcheck_int_of(Ma))) ) ).

tff(fact_4254_code__int_Oint__of__inject,axiom,
    ! [X_1: quickcheck_code_int,Y_2: quickcheck_code_int] :
      ( ( quickcheck_int_of(X_1) = quickcheck_int_of(Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_4255_Quickcheck__Narrowing_Oint__of__inject,axiom,
    ! [K_1: quickcheck_code_int,L: quickcheck_code_int] :
      ( ( quickcheck_int_of(K_1) = quickcheck_int_of(L) )
    <=> ( K_1 = L ) ) ).

tff(fact_4256_int__of__number,axiom,
    ! [K_2: int] : quickcheck_int_of(hAPP_i1732201573de_int(number1226105091de_int,K_2)) = hAPP_int_int(number_number_of_int,K_2) ).

tff(fact_4257_zero__code__int__code,axiom,
    zero_z891286103de_int = hAPP_i1732201573de_int(number1226105091de_int,pls) ).

tff(fact_4258_rcis__zero__arg,axiom,
    ! [R_3: real] : rcis(R_3,zero_zero_real) = of_real_complex(R_3) ).

tff(fact_4259_complex__mod__rcis,axiom,
    ! [R_3: real,A_3: real] : hAPP_complex_real(norm_norm_complex,rcis(R_3,A_3)) = hAPP_real_real(abs_abs_real,R_3) ).

tff(fact_4260_cis__rcis__eq,axiom,
    ! [A_3: real] : cis(A_3) = rcis(one_one_real,A_3) ).

tff(fact_4261_rcis__zero__mod,axiom,
    ! [A_3: real] : rcis(zero_zero_real,A_3) = zero_zero_complex ).

tff(fact_4262_Re__rcis,axiom,
    ! [R_3: real,A_3: real] : hAPP_complex_real(re,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3)) ).

tff(fact_4263_Im__rcis,axiom,
    ! [R_3: real,A_3: real] : hAPP_complex_real(im,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3)) ).

tff(fact_4264_rcis__mult,axiom,
    ! [R1: real,A_3: real,R2: real,B_2: real] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,rcis(R1,A_3)),rcis(R2,B_2)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) ).

tff(fact_4265_rcis__divide,axiom,
    ! [R1: real,A_3: real,R2: real,B_2: real] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,rcis(R1,A_3)),rcis(R2,B_2)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).

tff(fact_4266_rcis__def,axiom,
    ! [R_3: real,A_3: real] : rcis(R_3,A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),cis(A_3)) ).

tff(fact_4267_rcis__inverse,axiom,
    ! [R_3: real,A_3: real] : hAPP_complex_complex(invers1449016382omplex,rcis(R_3,A_3)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_3),hAPP_real_real(uminus_uminus_real,A_3)) ).

tff(fact_4268_div__mod__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] : quickc495462417de_int(N,M) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N),M)),hAPP_Q1762011733de_int(div_mo231679042de_int(N),M)) ).

tff(fact_4269_nat__of__def,axiom,
    ! [I: quickcheck_code_int] : quickcheck_nat_of(I) = hAPP_int_nat(nat_1,quickcheck_int_of(I)) ).

tff(fact_4270_around__zero_Opinduct,axiom,
    ! [P_1: fun_int_bool,A0: int] :
      ( hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),A0))
     => ( ! [I_2: int] :
            ( hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),I_2))
           => ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),zero_zero_int))
               => ( ( I_2 != zero_zero_int )
                 => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))) ) )
             => hBOOL(hAPP_int_bool(P_1,I_2)) ) )
       => hBOOL(hAPP_int_bool(P_1,A0)) ) ) ).

tff(fact_4271_fact__int__def,axiom,
    ! [X: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
       => ( hAPP_int_int(fact_fact_int,X) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat_1,X))) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
       => ( hAPP_int_int(fact_fact_int,X) = zero_zero_int ) ) ) ).

tff(fact_4272_fact__mono__int,axiom,
    ! [M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N))) ) ).

tff(fact_4273_fact__1__int,axiom,
    hAPP_int_int(fact_fact_int,one_one_int) = one_one_int ).

tff(fact_4274_fact__less__mono__int,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N))) ) ) ).

tff(fact_4275_fact__neg__int,axiom,
    ! [M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),zero_zero_int))
     => ( hAPP_int_int(fact_fact_int,M) = zero_zero_int ) ) ).

tff(fact_4276_fact__0__int,axiom,
    hAPP_int_int(fact_fact_int,zero_zero_int) = one_one_int ).

tff(fact_4277_transfer__nat__int__factorial__closure,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))) ) ).

tff(fact_4278_fact__nonzero__int,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_int_int(fact_fact_int,N) != zero_zero_int ) ) ).

tff(fact_4279_fact__ge__zero__int,axiom,
    ! [M: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,M))) ).

tff(fact_4280_transfer__int__nat__factorial,axiom,
    ! [X: nat] : hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X)) ).

tff(fact_4281_fact__gt__zero__int,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(fact_fact_int,N))) ) ).

tff(fact_4282_fact__ge__one__int,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(fact_fact_int,N))) ) ).

tff(fact_4283_fact__mono__int__aux,axiom,
    ! [M: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),K_2)))) ) ).

tff(fact_4284_dvd__fact__int,axiom,
    ! [N: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(fact_fact_int,N))) ) ) ).

tff(fact_4285_transfer__nat__int__factorial,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat_1,X)) = hAPP_int_nat(nat_1,hAPP_int_int(fact_fact_int,X)) ) ) ).

tff(fact_4286_fact__less__mono__int__aux,axiom,
    ! [M: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),one_one_int)),K_2)))) ) ) ).

tff(fact_4287_fact__plus__one__int,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
     => ( hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),hAPP_int_int(fact_fact_int,N)) ) ) ).

tff(fact_4288_fact__reduce__int,axiom,
    ! [N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( hAPP_int_int(fact_fact_int,N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))) ) ) ).

tff(fact_4289_length__around__zero,axiom,
    ! [I: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I))
     => ( hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat_1,I))),one_one_nat) ) ) ).

tff(fact_4290_Bnor__prime,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_5))
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),P_5)) = hAPP_int_nat(nat_1,A) ) ) ) ).

tff(fact_4291_lemma__BOLZANO,axiom,
    ! [A: real,B: real,P_1: fun_Pr1127657839l_bool] :
      ( ! [A_2: real,B_4: real,C_3: real] :
          ( ( hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
            & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3)))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) )
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) )
     => ( ! [X_2: real] :
          ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [A_2: real,B_4: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X_2))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B_4))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) )
               => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) ) )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))) ) ) ) ).

tff(fact_4292_Bnor__mem__zle__swap,axiom,
    ! [Ma: int,A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) ) ).

tff(fact_4293_Bnor__mem__zle,axiom,
    ! [B: int,A: int,Ma: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ).

tff(fact_4294_Bnor__fin,axiom,
    ! [A: int,Ma: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))) ).

tff(fact_4295_Bnor__mem__zg,axiom,
    ! [B: int,A: int,Ma: int] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) ) ).

tff(fact_4296_lemma__BOLZANO2,axiom,
    ! [P_1: fun_Pr1127657839l_bool] :
      ( ( ! [A_2: real,B_4: real,C_3: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3))
              & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4)))
              & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) )
           => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))) )
        & ! [X_2: real] :
          ? [D_2: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
            & ! [A_2: real,B_4: real] :
                ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X_2))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B_4))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) )
               => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) ) ) )
     => ! [A_2: real,B_4: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4))
         => hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) ) ) ).

tff(fact_4297_bezw__aux,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,Y))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).

tff(fact_4298_gcd__dvd2__nat,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),N)) ).

tff(fact_4299_gcd__dvd1__nat,axiom,
    ! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),M)) ).

tff(fact_4300_gcd__dvd__prod__nat,axiom,
    ! [M: nat,N: nat,K_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) ).

tff(fact_4301_coprime__exp2__nat,axiom,
    ! [N: nat,M: nat,A_3: nat,B_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),M)) = one_one_nat ) ) ).

tff(fact_4302_gcd__exp__nat,axiom,
    ! [A_3: nat,N: nat,B_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),N) ).

tff(fact_4303_gcd__semilattice__nat_Oinf_Oidem,axiom,
    ! [A_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),A_3) = A_3 ).

tff(fact_4304_gcd__idem__nat,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).

tff(fact_4305_gcd__commute__nat,axiom,
    ! [A_3: nat,B_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),A_3) ).

tff(fact_4306_gcd__semilattice__nat_Oinf__commute,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),X) ).

tff(fact_4307_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
    ! [A_3: nat,B_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) ).

tff(fact_4308_gcd__semilattice__nat_Oinf__left__idem,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) ).

tff(fact_4309_gcd__left__commute__nat,axiom,
    ! [B_2: nat,A_3: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) ).

tff(fact_4310_gcd__semilattice__nat_Oinf__left__commute,axiom,
    ! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Z_1)) ).

tff(fact_4311_gcd__assoc__nat,axiom,
    ! [A_3: nat,B_2: nat,C_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) ).

tff(fact_4312_gcd__semilattice__nat_Oinf__assoc,axiom,
    ! [X: nat,Y: nat,Z_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) ).

tff(fact_4313_gcd__semilattice__nat_Oless__infI2,axiom,
    ! [A_3: nat,B_2: nat,X: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).

tff(fact_4314_gcd__semilattice__nat_Oless__infI1,axiom,
    ! [B_2: nat,A_3: nat,X: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X))
        & ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).

tff(fact_4315_gcd__semilattice__nat_Ole__infE,axiom,
    ! [X: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) ) ) ).

tff(fact_4316_dvd__gcd__D2__nat,axiom,
    ! [K_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) ) ).

tff(fact_4317_dvd__gcd__D1__nat,axiom,
    ! [K_2: nat,M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ).

tff(fact_4318_gcd__semilattice__nat_Oinf__mono,axiom,
    ! [B_2: nat,D_3: nat,A_3: nat,C_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),D_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,C_1),D_3))) ) ) ).

tff(fact_4319_gcd__semilattice__nat_Oinf__greatest,axiom,
    ! [Z_1: nat,X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1))) ) ) ).

tff(fact_4320_gcd__semilattice__nat_Ole__infI,axiom,
    ! [B_2: nat,X: nat,A_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).

tff(fact_4321_gcd__greatest__nat,axiom,
    ! [N: nat,K_2: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) ) ) ).

tff(fact_4322_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ) ).

tff(fact_4323_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ) ).

tff(fact_4324_gcd__semilattice__nat_Ole__infI2,axiom,
    ! [A_3: nat,B_2: nat,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)) ) ).

tff(fact_4325_gcd__semilattice__nat_Ole__infI1,axiom,
    ! [B_2: nat,A_3: nat,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)) ) ).

tff(fact_4326_gcd__unique__nat,axiom,
    ! [B: nat,D: nat,A: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),A))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),B))
        & ! [E: nat] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),A))
              & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),B)) )
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D)) ) )
    <=> ( D = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

tff(fact_4327_gcd__semilattice__nat_Ole__inf__iff,axiom,
    ! [X_1: nat,Y_2: nat,Z_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_2),Z_2)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_2)) ) ) ).

tff(fact_4328_gcd__greatest__iff__nat,axiom,
    ! [K_1: nat,Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) ) ) ).

tff(fact_4329_gcd__semilattice__nat_Ole__iff__inf,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_2) = X_1 ) ) ).

tff(fact_4330_gcd__semilattice__nat_Oinf__le2,axiom,
    ! [X: nat,Y: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Y)) ).

tff(fact_4331_gcd__semilattice__nat_Oinf__le1,axiom,
    ! [X: nat,Y: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),X)) ).

tff(fact_4332_gcd__add2__nat,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).

tff(fact_4333_gcd__add1__nat,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).

tff(fact_4334_gcd__mult__distrib__nat,axiom,
    ! [K_2: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) ).

tff(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X) = one_one_nat ).

tff(fact_4336_gcd__1__nat,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

tff(fact_4337_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),one_one_nat) = one_one_nat ).

tff(fact_4338_gcd__0__left__nat,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

tff(fact_4339_gcd__0__nat,axiom,
    ! [X: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

tff(fact_4340_gcd__zero__nat,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N_1) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        & ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_4341_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_2) = zero_zero_nat )
    <=> ( ( X_1 = zero_zero_nat )
        & ( Y_2 = zero_zero_nat ) ) ) ).

tff(fact_4342_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ).

tff(fact_4343_gcd__Suc__0,axiom,
    ! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_4344_gcd__pos__nat,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N_1)))
    <=> ( ( Ma != zero_zero_nat )
        | ( N_1 != zero_zero_nat ) ) ) ).

tff(fact_4345_gcd__le1__nat,axiom,
    ! [B_2: nat,A_3: nat] :
      ( ( A_3 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),A_3)) ) ).

tff(fact_4346_gcd__le2__nat,axiom,
    ! [A_3: nat,B_2: nat] :
      ( ( B_2 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),B_2)) ) ).

tff(fact_4347_coprime__Suc__nat,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N)),N) = one_one_nat ).

tff(fact_4348_gcd__add__mult__nat,axiom,
    ! [M: nat,K_2: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).

tff(fact_4349_coprime__common__divisor__nat,axiom,
    ! [X: nat,A_3: nat,B_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2))
         => ( X = one_one_nat ) ) ) ) ).

tff(fact_4350_coprime__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat )
    <=> ! [D_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) )
        <=> ( D_2 = one_one_nat ) ) ) ).

tff(fact_4351_coprime__plus__one__nat,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N) = one_one_nat ).

tff(fact_4352_coprime__mul__eq__nat,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)) = one_one_nat )
    <=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),A) = one_one_nat )
        & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),B) = one_one_nat ) ) ) ).

tff(fact_4353_gcd__mult__cancel__nat,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

tff(fact_4354_coprime__mult__nat,axiom,
    ! [B_2: nat,D_3: nat,A_3: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat ) ) ) ).

tff(fact_4355_coprime__crossproduct__nat,axiom,
    ! [B: nat,C: nat,A: nat,D: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),D) = one_one_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B),C) = one_one_nat )
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D) )
        <=> ( ( A = B )
            & ( C = D ) ) ) ) ) ).

tff(fact_4356_coprime__lmult__nat,axiom,
    ! [D_3: nat,A_3: nat,B_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat ) ) ).

tff(fact_4357_coprime__rmult__nat,axiom,
    ! [D_3: nat,A_3: nat,B_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat ) ) ).

tff(fact_4358_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

tff(fact_4359_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

tff(fact_4360_gcd__nat_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ) ).

tff(fact_4361_gcd__non__0__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( Y != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ).

tff(fact_4362_coprime__exp__nat,axiom,
    ! [N: nat,D_3: nat,A_3: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)) = one_one_nat ) ) ).

tff(fact_4363_ceiling__real__of__nat,axiom,
    ! [N: nat] : archim856651990g_real(hAPP_nat_real(real_nat,N)) = hAPP_nat_int(semiri1621563631at_int,N) ).

tff(fact_4364_natceiling__def,axiom,
    ! [X: real] : natceiling(X) = hAPP_int_nat(nat_1,archim856651990g_real(X)) ).

tff(fact_4365_gcd__coprime__nat,axiom,
    ! [B_5: nat,A_5: nat,A_3: nat,B_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat )
     => ( ( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) )
       => ( ( B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) )
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_5),B_5) = one_one_nat ) ) ) ) ).

tff(fact_4366_coprime__minus__one__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N) = one_one_nat ) ) ).

tff(fact_4367_divides__mult__nat,axiom,
    ! [N: nat,M: nat,R_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_3))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = one_one_nat )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),R_3)) ) ) ) ).

tff(fact_4368_coprime__dvd__mult__iff__nat,axiom,
    ! [Ma: nat,K_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_1),N_1) = one_one_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) ) ) ).

tff(fact_4369_coprime__dvd__mult__nat,axiom,
    ! [M: nat,K_2: nat,N: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) = one_one_nat )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ) ).

tff(fact_4370_div__gcd__coprime__nat,axiom,
    ! [B_2: nat,A_3: nat] :
      ( ( ( A_3 != zero_zero_nat )
        | ( B_2 != zero_zero_nat ) )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) = one_one_nat ) ) ).

tff(fact_4371_invertible__coprime__nat,axiom,
    ! [X: nat,Y: nat,M: nat] :
      ( ( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),M) = one_one_nat )
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),M) = one_one_nat ) ) ).

tff(fact_4372_coprime__Suc__0__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat )
    <=> ! [D_2: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) )
        <=> ( D_2 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_4373_gcd__coprime__exists__nat,axiom,
    ! [A_3: nat,B_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat )
     => ? [A_4: nat,B_3: nat] :
          ( ( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) )
          & ( B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) )
          & ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_4),B_3) = one_one_nat ) ) ) ).

tff(fact_4374_bezout__nat,axiom,
    ! [B_2: nat,A_3: nat] :
      ( ( A_3 != zero_zero_nat )
     => ? [X_2: nat,Y_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) ) ).

tff(fact_4375_Cauchy__iff2,axiom,
    ! [X_3: fun_nat_real] :
      ( hBOOL(cauchy_real(X_3))
    <=> ! [J_1: nat] :
        ? [M_3: nat] :
        ! [M_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),M_1))
         => ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_2))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(X_3,M_1)),hAPP_nat_real(X_3,N_2)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))) ) ) ) ).

tff(fact_4376_ceiling__eq2,axiom,
    ! [N: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
       => ( archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

tff(fact_4377_ln__powr__bound,axiom,
    ! [A_3: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),A_3))) ) ) ).

tff(fact_4378_real__of__int__less__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) ) ).

tff(fact_4379_powr__not__zero,axiom,
    ! [X: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3) != zero_zero_real ).

tff(fact_4380_powr__one__eq__one,axiom,
    ! [A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_3) = one_one_real ).

tff(fact_4381_floor__real__of__int,axiom,
    ! [N: int] : archim1246769320r_real(hAPP_int_real(real_int,N)) = N ).

tff(fact_4382_real__of__int__floor__cancel,axiom,
    ! [X_1: real] :
      ( ( hAPP_int_real(real_int,archim1246769320r_real(X_1)) = X_1 )
    <=> ? [N_2: int] : X_1 = hAPP_int_real(real_int,N_2) ) ).

tff(fact_4383_powr__powr,axiom,
    ! [X: real,A_3: real,B_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)) ).

tff(fact_4384_powr__powr__swap,axiom,
    ! [X: real,A_3: real,B_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)),A_3) ).

tff(fact_4385_real__of__int__inject,axiom,
    ! [X_1: int,Y_2: int] :
      ( ( hAPP_int_real(real_int,X_1) = hAPP_int_real(real_int,Y_2) )
    <=> ( X_1 = Y_2 ) ) ).

tff(fact_4386_real__of__int__zero,axiom,
    hAPP_int_real(real_int,zero_zero_int) = zero_zero_real ).

tff(fact_4387_real__of__int__zero__cancel,axiom,
    ! [X_1: int] :
      ( ( hAPP_int_real(real_int,X_1) = zero_zero_real )
    <=> ( X_1 = zero_zero_int ) ) ).

tff(fact_4388_real__of__int__le__iff,axiom,
    ! [X_1: int,Y_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) ) ).

tff(fact_4389_real__of__int__diff,axiom,
    ! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).

tff(fact_4390_power__real__of__int,axiom,
    ! [X: int,N: nat] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) ).

tff(fact_4391_real__of__int__power,axiom,
    ! [X: int,N: nat] : hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) ).

tff(fact_4392_real__of__int__of__nat__eq,axiom,
    ! [N: nat] : hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_real(real_nat,N) ).

tff(fact_4393_real__of__int__add,axiom,
    ! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).

tff(fact_4394_real__number__of,axiom,
    ! [V_2: int] : hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_2)) = hAPP_int_real(number267125858f_real,V_2) ).

tff(fact_4395_real__of__one,axiom,
    hAPP_int_real(real_int,one_one_int) = one_one_real ).

tff(fact_4396_real__of__int__mult,axiom,
    ! [X: int,Y: int] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).

tff(fact_4397_ceiling__real__of__int,axiom,
    ! [N: int] : archim856651990g_real(hAPP_int_real(real_int,N)) = N ).

tff(fact_4398_real__of__int__ceiling__cancel,axiom,
    ! [X_1: real] :
      ( ( hAPP_int_real(real_int,archim856651990g_real(X_1)) = X_1 )
    <=> ? [N_2: int] : X_1 = hAPP_int_real(real_int,N_2) ) ).

tff(fact_4399_real__of__int__minus,axiom,
    ! [X: int] : hAPP_int_real(real_int,hAPP_int_int(uminus_uminus_int,X)) = hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,X)) ).

tff(fact_4400_real__of__int__abs,axiom,
    ! [X: int] : hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X)) = hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X)) ).

tff(fact_4401_real__of__int__floor__le,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),R_3)) ).

tff(fact_4402_real__of__int__ceiling__ge,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_int_real(real_int,archim856651990g_real(R_3)))) ).

tff(fact_4403_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y))) ).

tff(fact_4404_powr__gt__zero,axiom,
    ! [X: real,A_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))) ).

tff(fact_4405_powr__less__mono2,axiom,
    ! [Y: real,X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))) ) ) ) ).

tff(fact_4406_powr__less__mono2__neg,axiom,
    ! [Y: real,X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))) ) ) ) ).

tff(fact_4407_powr__mono,axiom,
    ! [X: real,A_3: real,B_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) ) ) ).

tff(fact_4408_powr__zero__eq__one,axiom,
    ! [X: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),zero_zero_real) = one_one_real ).

tff(fact_4409_powr__less__cancel__iff,axiom,
    ! [A: real,B: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) ) ) ).

tff(fact_4410_powr__less__mono,axiom,
    ! [X: real,A_3: real,B_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) ) ) ).

tff(fact_4411_powr__less__cancel,axiom,
    ! [X: real,A_3: real,B_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2)) ) ) ).

tff(fact_4412_powr__add,axiom,
    ! [X: real,A_3: real,B_2: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)) ).

tff(fact_4413_powr__divide2,axiom,
    ! [X: real,A_3: real,B_2: real] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).

tff(fact_4414_powr__minus,axiom,
    ! [X: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) ).

tff(fact_4415_real__of__int__div4,axiom,
    ! [N: int,X: int] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X)))) ).

tff(fact_4416_floor__less__eq,axiom,
    ! [X_1: real,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),A))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(real_int,A))) ) ).

tff(fact_4417_le__floor,axiom,
    ! [A_3: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X))) ) ).

tff(fact_4418_real__le__floor,axiom,
    ! [A_3: int,X: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X)) ) ).

tff(fact_4419_le__floor__eq,axiom,
    ! [A: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X_1)) ) ).

tff(fact_4420_floor__add,axiom,
    ! [X: real,A_3: int] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X)),A_3) ).

tff(fact_4421_less__ceiling__eq,axiom,
    ! [A: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,A)),X_1)) ) ).

tff(fact_4422_ceiling__le__eq,axiom,
    ! [X_1: real,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),A))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,A))) ) ).

tff(fact_4423_ceiling__le,axiom,
    ! [X: real,A_3: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3)) ) ).

tff(fact_4424_ceiling__le__real,axiom,
    ! [X: real,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3))) ) ).

tff(fact_4425_floor__power,axiom,
    ! [N: nat,X: real] :
      ( ( X = hAPP_int_real(real_int,archim1246769320r_real(X)) )
     => ( archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X)),N) ) ) ).

tff(fact_4426_floor__subtract,axiom,
    ! [X: real,A_3: int] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X)),A_3) ).

tff(fact_4427_ceiling__add,axiom,
    ! [X: real,A_3: int] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X)),A_3) ).

tff(fact_4428_floor__minus__real__of__int,axiom,
    ! [N: int] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N))) = hAPP_int_int(uminus_uminus_int,N) ).

tff(fact_4429_ceiling__subtract,axiom,
    ! [X: real,A_3: int] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X)),A_3) ).

tff(fact_4430_powr__mono2,axiom,
    ! [Y: real,X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))) ) ) ) ).

tff(fact_4431_powr__mult,axiom,
    ! [A_3: real,Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)) ) ) ) ).

tff(fact_4432_ge__one__powr__ge__zero,axiom,
    ! [A_3: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))) ) ) ).

tff(fact_4433_powr__le__cancel__iff,axiom,
    ! [A: real,B: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B)))
      <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) ) ) ).

tff(fact_4434_powr__one__gt__zero__iff,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),one_one_real) = X_1 )
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) ) ).

tff(fact_4435_powr__divide,axiom,
    ! [A_3: real,Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)) ) ) ) ).

tff(fact_4436_powr__minus__divide,axiom,
    ! [X: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) ).

tff(fact_4437_powr__def,axiom,
    ! [X: real,A_3: real] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3) = hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),hAPP_real_real(ln,X))) ).

tff(fact_4438_real__of__int__lt__zero__cancel__iff,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),zero_zero_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),zero_zero_int)) ) ).

tff(fact_4439_real__of__int__gt__zero__cancel__iff,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) ) ).

tff(fact_4440_real__of__int__ge__zero__cancel__iff,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) ) ).

tff(fact_4441_real__of__int__le__zero__cancel__iff,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),zero_zero_real))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int)) ) ).

tff(fact_4442_number__of__less__real__of__int__iff2,axiom,
    ! [Ma: int,N_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,N_1))) ) ).

tff(fact_4443_number__of__less__real__of__int__iff,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,N_1)),hAPP_int_real(real_int,Ma)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,N_1)),Ma)) ) ).

tff(fact_4444_number__of__le__real__of__int__iff2,axiom,
    ! [Ma: int,N_1: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N_1)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ma),hAPP_int_int(number_number_of_int,N_1))) ) ).

tff(fact_4445_number__of__le__real__of__int__iff,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N_1)),hAPP_int_real(real_int,Ma)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,N_1)),Ma)) ) ).

tff(fact_4446_int__less__real__le,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),Ma))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)),hAPP_int_real(real_int,Ma))) ) ).

tff(fact_4447_lemma__floor2,axiom,
    ! [N: int,X: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),one_one_real)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),X)) ) ).

tff(fact_4448_int__le__real__less,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),Ma))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Ma)),one_one_real))) ) ).

tff(fact_4449_real__nat__eq__real,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hAPP_nat_real(real_nat,hAPP_int_nat(nat_1,X)) = hAPP_int_real(real_int,X) ) ) ).

tff(fact_4450_real__of__int__div,axiom,
    ! [N: int,D_3: int] :
      ( ( D_3 != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N))
       => ( hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,D_3)) ) ) ) ).

tff(fact_4451_real__of__int__floor__add__one__ge,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real))) ).

tff(fact_4452_real__of__int__floor__add__one__gt,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real))) ).

tff(fact_4453_floor__eq,axiom,
    ! [N: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
       => ( archim1246769320r_real(X) = N ) ) ) ).

tff(fact_4454_tan__periodic__int,axiom,
    ! [X: real,I: int] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,I)),pi))) = hAPP_real_real(tan,X) ).

tff(fact_4455_real__of__int__floor__ge__diff__one,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3)))) ).

tff(fact_4456_real__of__int__floor__gt__diff__one,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3)))) ).

tff(fact_4457_real__of__int__ceiling__le__add__one,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),one_one_real))) ).

tff(fact_4458_real__of__int__ceiling__diff__one__le,axiom,
    ! [R_3: real] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),one_one_real)),R_3)) ).

tff(fact_4459_powr__realpow,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) ) ) ).

tff(fact_4460_ln__powr,axiom,
    ! [Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
       => ( hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(ln,X)) ) ) ) ).

tff(fact_4461_powr__log__cancel,axiom,
    ! [X: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
         => ( hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),hAPP_real_real(log(A_3),X)) = X ) ) ) ) ).

tff(fact_4462_log__powr__cancel,axiom,
    ! [Y: real,A_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
     => ( ( A_3 != one_one_real )
       => ( hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),Y)) = Y ) ) ) ).

tff(fact_4463_lemma__floor,axiom,
    ! [N: int,M: int,R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N)) ) ) ).

tff(fact_4464_real__of__int__div2,axiom,
    ! [N: int,X: int] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X))))) ).

tff(fact_4465_real__of__int__div3,axiom,
    ! [N: int,X: int] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X)))),one_one_real)) ).

tff(fact_4466_real__of__int__div__aux,axiom,
    ! [X: int,D_3: int] :
      ( ( D_3 != zero_zero_int )
     => ( hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X),D_3))),hAPP_int_real(real_int,D_3))) ) ) ).

tff(fact_4467_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
       => ( archim1246769320r_real(X) = N ) ) ) ).

tff(fact_4468_less__floor__eq,axiom,
    ! [A: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim1246769320r_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real)),X_1)) ) ).

tff(fact_4469_floor__le__eq,axiom,
    ! [X_1: real,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),A))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real))) ) ).

tff(fact_4470_ceiling__eq3,axiom,
    ! [N: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,N)),one_one_real)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N)))
       => ( archim856651990g_real(X) = N ) ) ) ).

tff(fact_4471_ceiling__less__eq,axiom,
    ! [X_1: real,A: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),A))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real))) ) ).

tff(fact_4472_le__ceiling__eq,axiom,
    ! [A: int,X_1: real] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim856651990g_real(X_1)))
    <=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real)),X_1)) ) ).

tff(fact_4473_log__powr,axiom,
    ! [B_2: real,Y: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( hAPP_real_real(log(B_2),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(log(B_2),X)) ) ) ) ).

tff(fact_4474_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
       => ( archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

tff(fact_4475_powr__realpow2,axiom,
    ! [N: nat,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( ( X = zero_zero_real )
           => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) = zero_zero_real ) )
          & ( ( X != zero_zero_real )
           => ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)) ) ) ) ) ) ).

tff(fact_4476_ln__powr__bound2,axiom,
    ! [A_3: real,X: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(ln,X)),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),A_3)),X))) ) ) ).

tff(fact_4477_reals__Archimedean__6b__int,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3))
     => ? [N_2: int] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ) ).

tff(fact_4478_reals__Archimedean__6c__int,axiom,
    ! [R_3: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real))
     => ? [N_2: int] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ) ).

tff(fact_4479_real__lb__ub__int,axiom,
    ! [R_3: real] :
    ? [N_2: int] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3))
      & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) ) ).

tff(fact_4480_coprime__pow,axiom,
    ! [C_1: nat,N: nat,A_3: nat,B_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,C_1),N) )
       => ? [R_1: nat,S_2: nat] :
            ( ( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N) )
            & ( B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_2),N) ) ) ) ) ).

tff(fact_4481_prime__power__mult,axiom,
    ! [X: nat,Y: nat,K_2: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_3))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) )
       => ? [I_2: nat,J_1: nat] :
            ( ( X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) )
            & ( Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) ) ) ) ) ).

tff(fact_4482_Euler__Fermat,axiom,
    ! [X: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(X),M) = one_one_int )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(phi,M))),one_one_int),M)) ) ) ).

tff(fact_4483_zgcd__zdvd1,axiom,
    ! [I: int,J_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),I)) ).

tff(fact_4484_zgcd__zdvd2,axiom,
    ! [I: int,J_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),J_2)) ).

tff(fact_4485_zgcd__zmult__cancel,axiom,
    ! [M: int,K_2: int,N: int] :
      ( ( hAPP_int_int(legacy_zgcd(K_2),N) = one_one_int )
     => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N) = hAPP_int_int(legacy_zgcd(M),N) ) ) ).

tff(fact_4486_zgcd__zgcd__zmult,axiom,
    ! [N: int,K_2: int,M: int] :
      ( ( hAPP_int_int(legacy_zgcd(K_2),M) = one_one_int )
     => ( ( hAPP_int_int(legacy_zgcd(N),M) = one_one_int )
       => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),M) = one_one_int ) ) ) ).

tff(fact_4487_zgcd__eq,axiom,
    ! [M: int,N: int] : hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)) ).

tff(fact_4488_zgcd__1,axiom,
    ! [M: int] : hAPP_int_int(legacy_zgcd(M),one_one_int) = one_one_int ).

tff(fact_4489_zgcd__1__left,axiom,
    ! [M: int] : hAPP_int_int(legacy_zgcd(one_one_int),M) = one_one_int ).

tff(fact_4490_zgcd0,axiom,
    ! [I_1: int,J: int] :
      ( ( hAPP_int_int(legacy_zgcd(I_1),J) = zero_zero_int )
    <=> ( ( I_1 = zero_zero_int )
        & ( J = zero_zero_int ) ) ) ).

tff(fact_4491_zgcd__greatest__iff,axiom,
    ! [K_1: int,Ma: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(legacy_zgcd(Ma),N_1)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) ) ) ).

tff(fact_4492_zgcd__greatest,axiom,
    ! [N: int,K_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M),N))) ) ) ).

tff(fact_4493_zgcd__assoc,axiom,
    ! [K_2: int,M: int,N: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M)),N) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N)) ).

tff(fact_4494_zgcd__left__commute,axiom,
    ! [K_2: int,M: int,N: int] : hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(M),hAPP_int_int(legacy_zgcd(K_2),N)) ).

tff(fact_4495_zgcd__commute,axiom,
    ! [I: int,J_2: int] : hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(J_2),I) ).

tff(fact_4496_zgcd__power__distrib,axiom,
    ! [A_3: int,B_2: int,N: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) ).

tff(fact_4497_zgcd__zminus,axiom,
    ! [I: int,J_2: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(uminus_uminus_int,I)),J_2) = hAPP_int_int(legacy_zgcd(I),J_2) ).

tff(fact_4498_zgcd__zminus2,axiom,
    ! [I: int,J_2: int] : hAPP_int_int(legacy_zgcd(I),hAPP_int_int(uminus_uminus_int,J_2)) = hAPP_int_int(legacy_zgcd(I),J_2) ).

tff(fact_4499_zgcd__self,axiom,
    ! [M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
     => ( hAPP_int_int(legacy_zgcd(M),M) = M ) ) ).

tff(fact_4500_zgcd__pos,axiom,
    ! [I: int,J_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(I),J_2))) ).

tff(fact_4501_zgcd__geq__zero,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X),Y))) ).

tff(fact_4502_zgcd__zadd__zmult,axiom,
    ! [M: int,N: int,K_2: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),K_2))),N) = hAPP_int_int(legacy_zgcd(M),N) ).

tff(fact_4503_zgcd__zmult__distrib2__abs,axiom,
    ! [K_2: int,M: int,N: int] : hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(legacy_zgcd(M),N)) ).

tff(fact_4504_zgcd__1__power__left__distrib,axiom,
    ! [N: nat,A_3: int,B_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int )
     => ( hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),B_2) = one_one_int ) ) ).

tff(fact_4505_zgcd__1__power__distrib,axiom,
    ! [N: nat,A_3: int,B_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int )
     => ( hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) = one_one_int ) ) ).

tff(fact_4506_zgcd__0__left,axiom,
    ! [M: int] : hAPP_int_int(legacy_zgcd(zero_zero_int),M) = hAPP_int_int(abs_abs_int,M) ).

tff(fact_4507_zgcd__0,axiom,
    ! [M: int] : hAPP_int_int(legacy_zgcd(M),zero_zero_int) = hAPP_int_int(abs_abs_int,M) ).

tff(fact_4508_zgcd__zdvd__zgcd__zmult,axiom,
    ! [M: int,N: int,K_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(M),N)),hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N))) ).

tff(fact_4509_zgcd__zmult__eq__self,axiom,
    ! [N: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) = K_2 ) ) ).

tff(fact_4510_zgcd__zmult__distrib2,axiom,
    ! [M: int,N: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) ) ) ).

tff(fact_4511_zgcd__zmult__eq__self2,axiom,
    ! [N: int,K_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2))
     => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),K_2) = K_2 ) ) ).

tff(fact_4512_zdvd__iff__zgcd,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),N_1))
      <=> ( hAPP_int_int(legacy_zgcd(N_1),Ma) = Ma ) ) ) ).

tff(fact_4513_div__zgcd__relprime,axiom,
    ! [B_2: int,A_3: int] :
      ( ( ( A_3 != zero_zero_int )
        | ( B_2 != zero_zero_int ) )
     => ( hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(legacy_zgcd(A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(legacy_zgcd(A_3),B_2))) = one_one_int ) ) ).

tff(fact_4514_zgcd__non__0,axiom,
    ! [M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)) ) ) ).

tff(fact_4515_zcong__zgcd__zmult__zmod,axiom,
    ! [N: int,A_3: int,B_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),N))
       => ( ( hAPP_int_int(legacy_zgcd(M),N) = one_one_int )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) ) ) ) ).

tff(fact_4516_zgcd__zmult__zdvd__zgcd,axiom,
    ! [M: int,K_2: int,N: int] :
      ( ( hAPP_int_int(legacy_zgcd(K_2),N) = one_one_int )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)),hAPP_int_int(legacy_zgcd(M),N))) ) ).

tff(fact_4517_zrelprime__zdvd__zmult,axiom,
    ! [M: int,N: int,K_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(N),K_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).

tff(fact_4518_zrelprime__dvd__mult,axiom,
    ! [K_2: int,I: int,J_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(I),J_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),K_2)) ) ) ).

tff(fact_4519_zgcd__0__1__iff,axiom,
    ! [Ma: int] :
      ( ( hAPP_int_int(legacy_zgcd(zero_zero_int),Ma) = one_one_int )
    <=> ( hAPP_int_int(abs_abs_int,Ma) = one_one_int ) ) ).

tff(fact_4520_zgcd__code,axiom,
    ! [K_1: int,L: int] : hAPP_int_int(legacy_zgcd(K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) ).

tff(fact_4521_zprime__imp__zrelprime,axiom,
    ! [N: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N))
       => ( hAPP_int_int(legacy_zgcd(N),P_3) = one_one_int ) ) ) ).

tff(fact_4522_zgcd1__iff__no__common__primedivisor,axiom,
    ! [A: int,B: int] :
      ( ( hAPP_int_int(legacy_zgcd(A),B) = one_one_int )
    <=> ~ ? [P_4: int] :
            ( hBOOL(hAPP_int_bool(zprime,P_4))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B)) ) ) ).

tff(fact_4523_zgcd__zcong__zgcd,axiom,
    ! [B_2: int,A_3: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( ( hAPP_int_int(legacy_zgcd(A_3),M) = one_one_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))
         => ( hAPP_int_int(legacy_zgcd(B_2),M) = one_one_int ) ) ) ) ).

tff(fact_4524_zless__zprime__imp__zrelprime,axiom,
    ! [N: int,P_3: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),P_3))
         => ( hAPP_int_int(legacy_zgcd(N),P_3) = one_one_int ) ) ) ) ).

tff(fact_4525_zcong__cancel,axiom,
    ! [A: int,B: int,K_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma))
     => ( ( hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),K_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),K_1)),Ma))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) ) ) ) ).

tff(fact_4526_zcong__cancel2,axiom,
    ! [A: int,B: int,K_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma))
     => ( ( hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),B)),Ma))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) ) ) ) ).

tff(fact_4527_zrelprime__zdvd__zmult__aux,axiom,
    ! [M: int,N: int,K_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(N),K_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ) ).

tff(fact_4528_Bnor__mem__if,axiom,
    ! [A: int,B: int,Ma: int] :
      ( ( hAPP_int_int(legacy_zgcd(B),Ma) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) ) ) ) ).

tff(fact_4529_int__relprime__odd__power__divisors,axiom,
    ! [A: int,B: int,C: int,X_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat_1,X_1)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat_1,X_1)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
         => ( ( hAPP_int_int(legacy_zgcd(A),B) = one_one_int )
           => ? [K: int] : A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat_1,X_1)) ) ) ) ) ).

tff(fact_4530_xzgcd__correct,axiom,
    ! [Ma: int,K_1: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( ( hAPP_int_int(legacy_zgcd(Ma),N_1) = K_1 )
      <=> ? [S_2: int,T: int] : xzgcd(Ma,N_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_2),T)) ) ) ).

tff(fact_4531_int__relprime__power__divisors,axiom,
    ! [A_3: int,B_2: int,C_1: int,N: nat] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C_1),N) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
       => ( ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int )
         => ? [K: int] : hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N) ) ) ) ).

tff(fact_4532_xzgcd__correct__aux2,axiom,
    ! [M: int,N: int,R_5: int,R_3: int,S_3: int,S_1: int,T_2: int,T_1: int,K_2: int] :
      ( ? [Sn: int,Tn: int] : xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
       => ( hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2 ) ) ) ).

tff(fact_4533_zcong__lineq__unique,axiom,
    ! [B_2: int,A_3: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( ( hAPP_int_int(legacy_zgcd(A_3),N) = one_one_int )
       => ? [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),N))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_2)),B_2),N))
            & ! [Y_1: int] :
                ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1))
                  & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),N))
                  & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Y_1)),B_2),N)) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_4534_zcong__lineq__ex,axiom,
    ! [A_3: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( ( hAPP_int_int(legacy_zgcd(A_3),N) = one_one_int )
       => ? [X_2: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_2)),one_one_int),N)) ) ) ).

tff(fact_4535_xzgcd__correct__aux1,axiom,
    ! [M: int,N: int,S_3: int,S_1: int,T_2: int,T_1: int,R_5: int,R_3: int,K_2: int] :
      ( ( hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2 )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3))
       => ? [Sn: int,Tn: int] : xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) ) ) ).

tff(fact_4536_int__triple__relprime__odd__power__divisors,axiom,
    ! [A: int,B: int,C: int,D: int,X_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),C) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D),hAPP_int_nat(nat_1,X_1)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat_1,X_1)))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))
         => ( ( hAPP_int_int(legacy_zgcd(A),B) = one_one_int )
           => ( ( hAPP_int_int(legacy_zgcd(B),C) = one_one_int )
             => ( ( hAPP_int_int(legacy_zgcd(C),A) = one_one_int )
               => ? [K: int,L_1: int,M_1: int] :
                    ( ( A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat_1,X_1)) )
                    & ( B = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat_1,X_1)) )
                    & ( C = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),hAPP_int_nat(nat_1,X_1)) ) ) ) ) ) ) ) ) ).

tff(fact_4537_zgcd__ex__linear,axiom,
    ! [M: int,K_2: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
     => ( ( hAPP_int_int(legacy_zgcd(M),N) = K_2 )
       => ? [S_2: int,T: int] : K_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T),N)) ) ) ).

tff(fact_4538_int__triple__relprime__power__divisors,axiom,
    ! [A_3: int,B_2: int,C_1: int,D_3: int,N: nat] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_3),N) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
       => ( ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int )
         => ( ( hAPP_int_int(legacy_zgcd(B_2),C_1) = one_one_int )
           => ( ( hAPP_int_int(legacy_zgcd(C_1),A_3) = one_one_int )
             => ? [K: int,L_1: int,M_1: int] :
                  ( ( hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N) )
                  & ( hAPP_int_int(abs_abs_int,B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N) )
                  & ( hAPP_int_int(abs_abs_int,C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),N) ) ) ) ) ) ) ) ).

tff(fact_4539_make__zrelprime,axiom,
    ! [B_2: int,A_3: int] :
      ( ( ( A_3 != zero_zero_int )
        | ( B_2 != zero_zero_int ) )
     => ? [C_3: int,D_2: int] :
          ( ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),C_3) )
          & ( B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),D_2) )
          & ( hAPP_int_int(legacy_zgcd(C_3),D_2) = one_one_int ) ) ) ).

tff(fact_4540_pair__lessI2,axiom,
    ! [Sa: nat,Ta: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Sa),Ta))
       => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less)) ) ) ).

tff(fact_4541_pair__lessI1,axiom,
    ! [Sa: nat,Ta: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
     => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less)) ) ).

tff(fact_4542_pair__leqI2,axiom,
    ! [Sa: nat,Ta: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Sa),Ta))
       => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq)) ) ) ).

tff(fact_4543_pair__leqI1,axiom,
    ! [Sa: nat,Ta: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
     => hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq)) ) ).

tff(fact_4544_polar__Ex,axiom,
    ! [Y: real,X: real] :
    ? [R_1: real,A_2: real] :
      ( ( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) )
      & ( Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ).

tff(fact_4545_dist__real__def,axiom,
    ! [X: real,Y: real] : hAPP_real_real(dist_dist_real(X),Y) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) ).

tff(fact_4546_dist__complex__def,axiom,
    ! [X: complex,Y: complex] : hAPP_complex_real(dist_dist_complex(X),Y) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) ).

tff(fact_4547_int__cases,axiom,
    ! [Z_1: int] :
      ( ! [N_2: nat] : Z_1 != hAPP_nat_int(semiri1621563631at_int,N_2)
     => ~ ! [N_2: nat] : Z_1 != hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_4548_int__of__nat__induct,axiom,
    ! [Z_2: int,P_1: fun_int_bool] :
      ( ! [N_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_2)))
     => ( ! [N_2: nat] : hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2)))))
       => hBOOL(hAPP_int_bool(P_1,Z_2)) ) ) ).

tff(fact_4549_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),zero_zero_real)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))) ) ) ).

tff(fact_4550_DERIV__nonneg__imp__nonincreasing,axiom,
    ! [F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => ? [Y_1: real] :
                ( hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) ) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,A)),hAPP_real_real(F,B))) ) ) ).

tff(fact_4551_gcd__nat__induct,axiom,
    ! [Ma: nat,N_1: nat,P_1: fun_nat_fun_nat_bool] :
      ( ! [M_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),zero_zero_nat))
     => ( ! [M_1: nat,N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
           => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),hAPP_nat_nat(div_mod_nat(M_1),N_2)))
             => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),N_2)) ) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Ma),N_1)) ) ) ).

tff(fact_4552_divides__fact,axiom,
    ! [N: nat,P_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),P_3))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_3),N))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N))) ) ) ).

tff(fact_4553_mod__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] : hAPP_Q1762011733de_int(div_mo231679042de_int(N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N)),quickcheck_int_of(M))) ).

tff(fact_4554_Quickcheck__Narrowing_Oof__int__inject,axiom,
    ! [N_1: int,Ma: int] :
      ( ( hAPP_i1732201573de_int(quickcheck_of_int,N_1) = hAPP_i1732201573de_int(quickcheck_of_int,Ma) )
    <=> ( N_1 = Ma ) ) ).

tff(fact_4555_int__of__inverse,axiom,
    ! [X: quickcheck_code_int] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X)) = X ).

tff(fact_4556_of__int__int__of,axiom,
    ! [K_2: quickcheck_code_int] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(K_2)) = K_2 ).

tff(fact_4557_int__of__of__int,axiom,
    ! [N: int] : quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N)) = N ).

tff(fact_4558_number__of__code__int__def,axiom,
    number1226105091de_int = quickcheck_of_int ).

tff(fact_4559_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact,M)),hAPP_nat_nat(fact,N))) ) ).

tff(fact_4560_times__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).

tff(fact_4561_plus__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).

tff(fact_4562_minus__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).

tff(fact_4563_zero__code__int__def,axiom,
    zero_z891286103de_int = hAPP_i1732201573de_int(quickcheck_of_int,zero_zero_int) ).

tff(fact_4564_fact__lt,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N))) ).

tff(fact_4565_fact_Osimps_I1_J,axiom,
    hAPP_nat_nat(fact,zero_zero_nat) = one_one_nat ).

tff(fact_4566_fact_Osimps_I2_J,axiom,
    ! [N: nat] : hAPP_nat_nat(fact,hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(fact,N)) ).

tff(fact_4567_fact__le,axiom,
    ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N))) ).

tff(fact_4568_one__code__int__def,axiom,
    one_on1684967323de_int = hAPP_i1732201573de_int(quickcheck_of_int,one_one_int) ).

tff(fact_4569_div__code__int__def,axiom,
    ! [N: quickcheck_code_int,M: quickcheck_code_int] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).

tff(fact_4570_euclid__bound,axiom,
    ! [N: nat] :
    ? [P_4: nat] :
      ( hBOOL(hAPP_nat_bool(prime,P_4))
      & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),P_4))
      & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N)))) ) ).

tff(fact_4571_weak__decr__stable,axiom,
    ! [F: fun_nat_nat] :
      ( ! [I_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_2))),hAPP_nat_nat(F,I_2)))
     => ? [I_2: nat] :
        ! [K: nat] : hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)) = hAPP_nat_nat(F,I_2) ) ).

tff(fact_4572_norm__frac_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [A_2: int,B_4: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)))
           => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),zero_zero_int))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_4))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4573_upto_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [I_2: int,J_1: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,I_2),J_1)))
           => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)),J_1)) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4574_small__lazy_H_Opinduct,axiom,
    ! [P_1: fun_int_fun_int_bool,A0: int,A1: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1)))
     => ( ! [D_2: int,I_2: int] :
            ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,D_2),I_2)))
           => ( ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,D_2),I_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))) )
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),I_2)) ) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4575_norRRset__def,axiom,
    ! [Ma: int] : hAPP_i1948725293t_bool(norRRset,Ma) = hAPP_i1948725293t_bool(bnorRset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ma),one_one_int)),Ma) ).

tff(fact_4576_real__eq__of__int,axiom,
    real_int = ring_1_of_int_real ).

tff(fact_4577_real__of__int__def,axiom,
    real_int = ring_1_of_int_real ).

tff(fact_4578_complex__Im__of__int,axiom,
    ! [Z_1: int] : hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1)) = zero_zero_real ).

tff(fact_4579_complex__Re__of__int,axiom,
    ! [Z_1: int] : hAPP_complex_real(re,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_real(ring_1_of_int_real,Z_1) ).

tff(fact_4580_complex__cnj__of__int,axiom,
    ! [Z_1: int] : hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_complex(ring_11397209091omplex,Z_1) ).

tff(fact_4581_of__int__int__eq,axiom,
    ! [N: nat] : hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,N) ).

tff(fact_4582_number__of__real__def,axiom,
    ! [X: int] : hAPP_int_real(number267125858f_real,X) = hAPP_int_real(ring_1_of_int_real,X) ).

tff(fact_4583_int__number__of__def,axiom,
    ! [W: int] : hAPP_int_int(number_number_of_int,W) = hAPP_int_int(ring_1_of_int_int,W) ).

tff(fact_4584_complex__number__of__def,axiom,
    ! [W: int] : hAPP_int_complex(number528085621omplex,W) = hAPP_int_complex(ring_11397209091omplex,W) ).

tff(fact_4585_phi__def,axiom,
    ! [Ma: int] : hAPP_int_nat(phi,Ma) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) ).

tff(fact_4586_norR__mem__unique,axiom,
    ! [A: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( ( hAPP_int_int(legacy_zgcd(A),Ma) = one_one_int )
       => ? [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),X_2),Ma))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(norRRset,Ma)))
            & ! [Y_1: int] :
                ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),Y_1),Ma))
                  & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_1),hAPP_i1948725293t_bool(norRRset,Ma))) )
               => ( Y_1 = X_2 ) ) ) ) ) ).

tff(fact_4587_card__nor__eq__noX,axiom,
    ! [X_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( ( hAPP_int_int(legacy_zgcd(X_1),Ma) = one_one_int )
       => ( hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(Ma),X_1)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) ) ) ) ).

tff(fact_4588_noX__is__RRset,axiom,
    ! [X_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( ( hAPP_int_int(legacy_zgcd(X_1),Ma) = one_one_int )
       => hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(Ma),X_1)),Ma)) ) ) ).

tff(fact_4589_RRset__gcd,axiom,
    ! [A: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
       => ( hAPP_int_int(legacy_zgcd(A),Ma) = one_one_int ) ) ) ).

tff(fact_4590_RRset__zcong__eq,axiom,
    ! [A: int,B: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
           => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),A_1))
             => ( A = B ) ) ) ) ) ) ).

tff(fact_4591_RRset2norRR__correct,axiom,
    ! [A: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ) ).

tff(fact_4592_RRset2norRR__correct2,axiom,
    ! [A: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ).

tff(fact_4593_RRset2norRR__correct1,axiom,
    ! [A: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma)) ) ) ) ).

tff(fact_4594_is__RRset__def,axiom,
    ! [A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
    <=> ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma)))
        & ( hAPP_f957591787ol_nat(finite_card_int,A_1) = hAPP_int_nat(phi,Ma) ) ) ) ).

tff(fact_4595_RsetR__fin,axiom,
    ! [A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma)))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1)) ) ).

tff(fact_4596_Bnor__in__RsetR,axiom,
    ! [A: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma))
     => hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(A),Ma)),rsetR(Ma))) ) ).

tff(fact_4597_RRset2norRR__eq__norR,axiom,
    ! [A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
       => ( image_int_int(rRset2norRR(A_1,Ma),A_1) = hAPP_i1948725293t_bool(norRRset,Ma) ) ) ) ).

tff(fact_4598_Nitpick_Oint__lcm__def,axiom,
    ! [X: int,Y: int] : hAPP_int_int(int_lcm(X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Y)))) ).

tff(fact_4599_ResSet__image,axiom,
    ! [F: fun_int_int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),A_1))
       => ( ! [X_2: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1))
             => ! [Xa: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1))
                 => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X_2)),hAPP_int_int(F,Xa)),Ma))
                   => ( X_2 = Xa ) ) ) )
         => hBOOL(hAPP_f448129468l_bool(resSet(Ma),image_int_int(F,A_1))) ) ) ) ).

tff(fact_4600_transfer__nat__int__set__relations_I5_J,axiom,
    ! [A_1: fun_nat_bool,B_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) ) ).

tff(fact_4601_transfer__nat__int__set__relations_I4_J,axiom,
    ! [A_1: fun_nat_bool,B_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) ) ).

tff(fact_4602_transfer__nat__int__set__relations_I3_J,axiom,
    ! [A_1: fun_nat_bool,B_1: fun_nat_bool] :
      ( ( A_1 = B_1 )
    <=> ( image_nat_int(semiri1621563631at_int,A_1) = image_nat_int(semiri1621563631at_int,B_1) ) ) ).

tff(fact_4603_transfer__nat__int__set__relations_I2_J,axiom,
    ! [X_1: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A_1))) ) ).

tff(fact_4604_transfer__int__nat__set__return__embed,axiom,
    ! [A_1: fun_nat_bool] : image_int_nat(nat_1,image_nat_int(semiri1621563631at_int,A_1)) = A_1 ).

tff(fact_4605_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [A_1: fun_nat_bool] : hAPP_f22106695ol_nat(finite_card_nat,A_1) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A_1)) ).

tff(fact_4606_transfer__nat__int__set__relations_I1_J,axiom,
    ! [A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A_1))) ) ).

tff(fact_4607_SetS__def,axiom,
    ! [A: int,P_5: int] : setS(A,P_5) = image_275383677t_bool(multInvPair(A,P_5),hAPP_i1948725293t_bool(sRStar,P_5)) ).

tff(fact_4608_infinite__nat__iff__unbounded__le,axiom,
    ! [S: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ! [M_1: nat] :
        ? [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
          & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) ) ) ).

tff(fact_4609_infinite__nat__iff__unbounded,axiom,
    ! [S: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ! [M_1: nat] :
        ? [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2))
          & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) ) ) ).

tff(fact_4610_Nitpick_Onat__lcm__def,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(nat_lcm(X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(nat_gcd(X),Y)) ).

tff(fact_4611_transfer__int__nat__set__relations_I4_J,axiom,
    ! [B_1: fun_int_bool,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1))
        <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat_1,A_1)),image_int_nat(nat_1,B_1))) ) ) ) ).

tff(fact_4612_gcd__eq__nitpick__gcd,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(nat_gcd(X),Y) ).

tff(fact_4613_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
    ! [X_1: int,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ) ).

tff(fact_4614_nat__set__def,axiom,
    ! [S: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,S))
    <=> ! [X_2: int] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),S))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) ) ) ).

tff(fact_4615_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,
    ! [C_2: fun_nat_bool] : hBOOL(hAPP_f448129468l_bool(nat_nat_set,image_nat_int(semiri1621563631at_int,C_2))) ).

tff(fact_4616_transfer__int__nat__set__relations_I3_J,axiom,
    ! [B_1: fun_int_bool,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1))
       => ( ( A_1 = B_1 )
        <=> ( image_int_nat(nat_1,A_1) = image_int_nat(nat_1,B_1) ) ) ) ) ).

tff(fact_4617_nat__gcd_Osimps,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( hAPP_nat_nat(nat_gcd(X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( hAPP_nat_nat(nat_gcd(X),Y) = hAPP_nat_nat(nat_gcd(Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ) ).

tff(fact_4618_transfer__nat__int__set__return__embed,axiom,
    ! [A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( image_nat_int(semiri1621563631at_int,image_int_nat(nat_1,A_1)) = A_1 ) ) ).

tff(fact_4619_transfer__int__nat__set__relations_I1_J,axiom,
    ! [A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1))
      <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat_1,A_1))) ) ) ).

tff(fact_4620_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
    ! [A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hAPP_f957591787ol_nat(finite_card_int,A_1) = hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat_1,A_1)) ) ) ).

tff(fact_4621_transfer__int__nat__set__relations_I5_J,axiom,
    ! [B_1: fun_int_bool,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1))
        <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat_1,A_1)),image_int_nat(nat_1,B_1))) ) ) ) ).

tff(fact_4622_nat__gcd_Opsimps,axiom,
    ! [X_1: nat,Y_2: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_1),Y_2)))
     => ( ( ( Y_2 = zero_zero_nat )
         => ( hAPP_nat_nat(nat_gcd(X_1),Y_2) = X_1 ) )
        & ( ( Y_2 != zero_zero_nat )
         => ( hAPP_nat_nat(nat_gcd(X_1),Y_2) = hAPP_nat_nat(nat_gcd(Y_2),hAPP_nat_nat(div_mod_nat(X_1),Y_2)) ) ) ) ) ).

tff(fact_4623_Nitpick_Oint__gcd__def,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(int_gcd,X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_gcd(hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Y)))) ).

tff(fact_4624_nat__gcd_Opinduct,axiom,
    ! [P_1: fun_nat_fun_nat_bool,A0: nat,A1: nat] :
      ( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A0),A1)))
     => ( ! [X_2: nat,Y_1: nat] :
            ( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_2),Y_1)))
           => ( ( ( Y_1 != zero_zero_nat )
               => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y_1),hAPP_nat_nat(div_mod_nat(X_2),Y_1))) )
             => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X_2),Y_1)) ) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1)) ) ) ).

tff(fact_4625_finite__nat__set__iff__bounded__le,axiom,
    ! [N_4: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4))
    <=> ? [M_1: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_1)) ) ) ).

tff(fact_4626_finite__nat__set__iff__bounded,axiom,
    ! [N_4: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4))
    <=> ? [M_1: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_1)) ) ) ).

tff(fact_4627_bounded__nat__set__is__finite,axiom,
    ! [N_1: nat,N_4: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),N_1)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)) ) ).

tff(fact_4628_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

tff(fact_4629_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U))) ).

tff(fact_4630_Rats__abs__nat__div__natE,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real))
     => ~ ! [M_1: nat,N_2: nat] :
            ( ( N_2 != zero_zero_nat )
           => ( ( hAPP_real_real(abs_abs_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_2)) )
             => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_2) != one_one_nat ) ) ) ) ).

tff(fact_4631_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U))) ).

tff(fact_4632_Rats__real__nat,axiom,
    ! [N_1: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N_1)),field_1210416355s_real)) ).

tff(fact_4633_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),hAPP_nat_nat(suc,L)) ).

tff(fact_4634_Rats__dense__in__nn__real,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
       => ? [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),X_2))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ) ).

tff(fact_4635_DERIV__isconst3,axiom,
    ! [F: fun_real_real,Y_2: real,X_1: real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B)))
       => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Y_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B)))
         => ( ! [X_2: real] :
                ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B)))
               => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
           => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_2) ) ) ) ) ) ).

tff(fact_4636_iszero__rat,axiom,
    ! [K_1: int] :
      ( hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K_1)))
    <=> hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K_1))) ) ).

tff(fact_4637_Rats__dense__in__real,axiom,
    ! [X_1: real,Y_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))
     => ? [X_2: real] :
          ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),X_2))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) ) ) ).

tff(fact_4638_Sup__greaterThanLessThan,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
     => ( comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Y_2),X_1)) = X_1 ) ) ).

tff(fact_4639_Sup__upper__EX,axiom,
    ! [X_1: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3))
     => ( ? [Z: real] :
          ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z)) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),comple124823625p_real(X_3))) ) ) ).

tff(fact_4640_SupInf_OSup__upper,axiom,
    ! [Z_2: real,X_1: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3))
     => ( ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),comple124823625p_real(X_3))) ) ) ).

tff(fact_4641_Frct__code__post_I4_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int)) = hAPP_int_rat(number_number_of_rat,K_2) ).

tff(fact_4642_Frct__code__post_I2_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,K_2),zero_zero_int)) = zero_zero_rat ).

tff(fact_4643_Frct__code__post_I1_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_2)) = zero_zero_rat ).

tff(fact_4644_Frct__code__post_I3_J,axiom,
    frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

tff(fact_4645_Frct__code__post_I6_J,axiom,
    ! [K_2: int,L_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) ).

tff(fact_4646_Frct__code__post_I5_J,axiom,
    ! [K_2: int] : frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),hAPP_int_int(number_number_of_int,K_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) ).

tff(fact_4647_Sup__eq__maximum,axiom,
    ! [Z_2: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_3))
     => ( ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)) )
       => ( comple124823625p_real(X_3) = Z_2 ) ) ) ).

tff(fact_4648_SupInf_OSup__upper2,axiom,
    ! [Z_2: real,Y_2: real,X_1: real,X_3: fun_real_bool] :
      ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1))
       => ( ! [X_2: real] :
              ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)) )
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),comple124823625p_real(X_3))) ) ) ) ).

tff(fact_4649_abs__rat__def,axiom,
    ! [Q_2: rat] :
      ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,Q_2) = hAPP_rat_rat(uminus_uminus_rat,Q_2) ) )
      & ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat))
       => ( hAPP_rat_rat(abs_abs_rat,Q_2) = Q_2 ) ) ) ).

tff(fact_4650_sgn__rat__def,axiom,
    ! [Q_2: rat] :
      ( ( ( Q_2 = zero_zero_rat )
       => ( hAPP_rat_rat(sgn_sgn_rat,Q_2) = zero_zero_rat ) )
      & ( ( Q_2 != zero_zero_rat )
       => ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2))
           => ( hAPP_rat_rat(sgn_sgn_rat,Q_2) = one_one_rat ) )
          & ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2))
           => ( hAPP_rat_rat(sgn_sgn_rat,Q_2) = hAPP_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

tff(fact_4651_divide__rat__def,axiom,
    ! [Q_2: rat,R_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_2),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_2),hAPP_rat_rat(inverse_inverse_rat,R_3)) ).

tff(fact_4652_normalize__negative,axiom,
    ! [P_3: int,Q_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_2),zero_zero_int))
     => ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(uminus_uminus_int,P_3)),hAPP_int_int(uminus_uminus_int,Q_2))) ) ) ).

tff(fact_4653_rat__sgn__code,axiom,
    ! [P_3: rat] : quotient_of(hAPP_rat_rat(sgn_sgn_rat,P_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(sgn_sgn_int,hAPP_P1175774780nt_int(product_fst_int_int,quotient_of(P_3)))),one_one_int) ).

tff(fact_4654_quotient__of__inject,axiom,
    ! [A_3: rat,B_2: rat] :
      ( ( quotient_of(A_3) = quotient_of(B_2) )
     => ( A_3 = B_2 ) ) ).

tff(fact_4655_quotient__of__inject__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( quotient_of(A) = quotient_of(B) )
    <=> ( A = B ) ) ).

tff(fact_4656_diff__rat__def,axiom,
    ! [Q_2: rat,R_3: rat] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_2),R_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_2),hAPP_rat_rat(uminus_uminus_rat,R_3)) ).

tff(fact_4657_less__rat__def,axiom,
    ! [Z_2: rat,W_1: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),W_1))
    <=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),W_1))
        & ( Z_2 != W_1 ) ) ) ).

tff(fact_4658_quotient__of__denom__pos,axiom,
    ! [R_3: rat,P_3: int,Q_2: int] :
      ( ( quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)) ) ).

tff(fact_4659_rat__one__code,axiom,
    quotient_of(one_one_rat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

tff(fact_4660_normalize__denom__pos,axiom,
    ! [R_3: product_prod_int_int,P_3: int,Q_2: int] :
      ( ( normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) )
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)) ) ).

tff(fact_4661_normalize__denom__zero,axiom,
    ! [P_3: int] : normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),zero_zero_int)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

tff(fact_4662_normalize__crossproduct,axiom,
    ! [P_3: int,R_3: int,S_1: int,Q_2: int] :
      ( ( Q_2 != zero_zero_int )
     => ( ( S_1 != zero_zero_int )
       => ( ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_3),S_1)) )
         => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,P_3),S_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),Q_2) ) ) ) ) ).

tff(fact_4663_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

tff(fact_4664_quotient__of__number_I3_J,axiom,
    ! [K_2: int] : quotient_of(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int) ).

tff(fact_4665_obtain__pos__sum,axiom,
    ! [R_3: rat] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_3))
     => ~ ! [S_2: rat] :
            ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),S_2))
           => ! [T: rat] :
                ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),T))
               => ( R_3 != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T) ) ) ) ) ).

tff(fact_4666_ratrel__iff,axiom,
    ! [X_1: product_prod_int_int,Y_2: product_prod_int_int] :
      ( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_1),Y_2)),ratrel))
    <=> ( ( hAPP_P1175774780nt_int(product_snd_int_int,X_1) != zero_zero_int )
        & ( hAPP_P1175774780nt_int(product_snd_int_int,Y_2) != zero_zero_int )
        & ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_1)) ) ) ) ).

tff(fact_4667_Fract__1__number__of,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),hAPP_int_int(number_number_of_int,K_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) ).

tff(fact_4668_Ratreal__number__of__quotient2,axiom,
    ! [R_3: int,S_1: int] : ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_3)),hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) ).

tff(fact_4669_quotient__of__eq,axiom,
    ! [A_3: int,B_2: int,P_3: int,Q_2: int] :
      ( ( quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ) ).

tff(fact_4670_eq__rat_I2_J,axiom,
    ! [A_3: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),zero_zero_int) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

tff(fact_4671_one__rat,axiom,
    one_one_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) ).

tff(fact_4672_rat__number__collapse_I2_J,axiom,
    hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) = one_one_rat ).

tff(fact_4673_rat__number__collapse_I1_J,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),K_2) = zero_zero_rat ).

tff(fact_4674_rat__number__collapse_I4_J,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),zero_zero_int) = zero_zero_rat ).

tff(fact_4675_eq__rat_I3_J,axiom,
    ! [A_3: int,C_1: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),A_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),C_1) ).

tff(fact_4676_of__nat__rat,axiom,
    ! [K_2: nat] : hAPP_nat_rat(semiri151668891at_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) ).

tff(fact_4677_Fract__of__nat__eq,axiom,
    ! [K_2: nat] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) = hAPP_nat_rat(semiri151668891at_rat,K_2) ).

tff(fact_4678_of__int__rat,axiom,
    ! [K_2: int] : hAPP_int_rat(ring_1_of_int_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) ).

tff(fact_4679_Fract__of__int__eq,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) = hAPP_int_rat(ring_1_of_int_rat,K_2) ).

tff(fact_4680_Fract__of__int__quotient,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),L_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(ring_1_of_int_rat,K_2)),hAPP_int_rat(ring_1_of_int_rat,L_2)) ).

tff(fact_4681_rat__number__of__def,axiom,
    ! [W: int] : hAPP_int_rat(number_number_of_rat,W) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,W),one_one_int) ).

tff(fact_4682_real__floor__code,axiom,
    ! [X: rat] : archim1246769320r_real(ratreal(X)) = archim791455193or_rat(X) ).

tff(fact_4683_floor__Fract,axiom,
    ! [A_3: int,B_2: int] : archim791455193or_rat(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) ).

tff(fact_4684_inverse__rat,axiom,
    ! [A_3: int,B_2: int] : hAPP_rat_rat(inverse_inverse_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,B_2),A_3) ).

tff(fact_4685_minus__rat__cancel,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ).

tff(fact_4686_mult__rat__cancel,axiom,
    ! [A_3: int,B_2: int,C_1: int] :
      ( ( C_1 != zero_zero_int )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ) ).

tff(fact_4687_eq__rat_I1_J,axiom,
    ! [A: int,C: int,D: int,B: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D) )
        <=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

tff(fact_4688_divide__rat,axiom,
    ! [A_3: int,B_2: int,C_1: int,D_3: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) ).

tff(fact_4689_minus__rat,axiom,
    ! [A_3: int,B_2: int] : hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),B_2) ).

tff(fact_4690_mult__rat,axiom,
    ! [A_3: int,B_2: int,C_1: int,D_3: int] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) ).

tff(fact_4691_abs__rat,axiom,
    ! [A_3: int,B_2: int] : hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_int(abs_abs_int,B_2)) ).

tff(fact_4692_normalize__eq,axiom,
    ! [A_3: int,B_2: int,P_3: int,Q_2: int] :
      ( ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ) ).

tff(fact_4693_real__less__code,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(X_1)),ratreal(Y_2)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2)) ) ).

tff(fact_4694_Ratreal__number__collapse_I1_J,axiom,
    ratreal(zero_zero_rat) = zero_zero_real ).

tff(fact_4695_zero__real__code,axiom,
    zero_zero_real = ratreal(zero_zero_rat) ).

tff(fact_4696_number__of__real__code,axiom,
    ! [K_2: int] : hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2)) ).

tff(fact_4697_Ratreal__number__collapse_I3_J,axiom,
    ! [K_2: int] : ratreal(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_int_real(number267125858f_real,K_2) ).

tff(fact_4698_Ratreal__number__collapse_I2_J,axiom,
    ratreal(one_one_rat) = one_one_real ).

tff(fact_4699_one__real__code,axiom,
    one_one_real = ratreal(one_one_rat) ).

tff(fact_4700_real__less__eq__code,axiom,
    ! [X_1: rat,Y_2: rat] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(X_1)),ratreal(Y_2)))
    <=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2)) ) ).

tff(fact_4701_real__plus__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Y)) ).

tff(fact_4702_real__minus__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),Y)) ).

tff(fact_4703_real__uminus__code,axiom,
    ! [X: rat] : hAPP_real_real(uminus_uminus_real,ratreal(X)) = ratreal(hAPP_rat_rat(uminus_uminus_rat,X)) ).

tff(fact_4704_real__divide__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X),Y)) ).

tff(fact_4705_real__times__code,axiom,
    ! [X: rat,Y: rat] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X),Y)) ).

tff(fact_4706_real__inverse__code,axiom,
    ! [X: rat] : hAPP_real_real(inverse_inverse_real,ratreal(X)) = ratreal(hAPP_rat_rat(inverse_inverse_rat,X)) ).

tff(fact_4707_zero__rat,axiom,
    zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

tff(fact_4708_rat__number__expand_I3_J,axiom,
    ! [K_2: int] : hAPP_int_rat(number_number_of_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) ).

tff(fact_4709_rat__number__collapse_I3_J,axiom,
    ! [K_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = hAPP_int_rat(number_number_of_rat,K_2) ).

tff(fact_4710_Fract__number__of__quotient,axiom,
    ! [K_2: int,L_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2)) ).

tff(fact_4711_quotient__of__Fract,axiom,
    ! [A_3: int,B_2: int] : quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) ).

tff(fact_4712_Frct__def,axiom,
    ! [P_3: product_prod_int_int] : frct(P_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)) ).

tff(fact_4713_less__rat,axiom,
    ! [A: int,C: int,D: int,B: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

tff(fact_4714_Ratreal__number__of__quotient,axiom,
    ! [R_3: int,S_1: int] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_3))),ratreal(hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) ).

tff(fact_4715_sgn__rat,axiom,
    ! [A_3: int,B_2: int] : hAPP_rat_rat(sgn_sgn_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,A_3)),hAPP_int_int(sgn_sgn_int,B_2))) ).

tff(fact_4716_Fract__less__zero__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) ) ) ).

tff(fact_4717_zero__less__Fract__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) ) ) ).

tff(fact_4718_le__rat,axiom,
    ! [A: int,C: int,D: int,B: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D)))
        <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).

tff(fact_4719_add__rat,axiom,
    ! [A_3: int,C_1: int,D_3: int,B_2: int] :
      ( ( B_2 != zero_zero_int )
     => ( ( D_3 != zero_zero_int )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) ) ) ) ).

tff(fact_4720_diff__rat,axiom,
    ! [A_3: int,C_1: int,D_3: int,B_2: int] :
      ( ( B_2 != zero_zero_int )
     => ( ( D_3 != zero_zero_int )
       => ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) ) ) ) ).

tff(fact_4721_Fract__less__one__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) ) ) ).

tff(fact_4722_one__less__Fract__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) ) ) ).

tff(fact_4723_Fract__add__one,axiom,
    ! [M: int,N: int] :
      ( ( N != zero_zero_int )
     => ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)),N) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,M),N)),one_one_rat) ) ) ).

tff(fact_4724_zero__le__Fract__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

tff(fact_4725_Fract__le__zero__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

tff(fact_4726_rat__floor__lemma,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)))
      & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),one_one_int)))) ) ).

tff(fact_4727_one__le__Fract__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) ) ) ).

tff(fact_4728_Fract__le__one__iff,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) ) ) ).

tff(fact_4729_Rat__induct__pos,axiom,
    ! [Q_1: rat,P_1: fun_rat_bool] :
      ( ! [A_2: int,B_4: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
         => hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))) )
     => hBOOL(hAPP_rat_bool(P_1,Q_1)) ) ).

tff(fact_4730_adjust__eq,axiom,
    ! [B: int,Q_1: int,R_2: int] : hAPP_P1975530577nt_int(adjust(B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)) = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),R_2)) ).

tff(fact_4731_finite__Collect__less__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1)))) ).

tff(fact_4732_finite__Collect__le__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1)))) ).

tff(fact_4733_transfer__morphism__int__nat,axiom,
    hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(semiri1621563631at_int),cOMBK_bool_nat(fTrue))) ).

tff(fact_4734_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
    ! [P_1: fun_int_bool] : collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = image_nat_int(semiri1621563631at_int,collect_nat(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P_1),semiri1621563631at_int))) ).

tff(fact_4735_int__card__bdd__int__set__l__le,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1))))) = N_1 ) ) ).

tff(fact_4736_card__bdd__int__set__l,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1)))) = hAPP_int_nat(nat_1,N_1) ) ) ).

tff(fact_4737_card__bdd__int__set__l__le,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))) = hAPP_int_nat(nat_1,N_1) ) ) ).

tff(fact_4738_bdd__int__set__l__le__finite,axiom,
    ! [N_1: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1))))) ).

tff(fact_4739_bdd__int__set__l__finite,axiom,
    ! [N_1: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1))))) ).

tff(fact_4740_DERIV__cos__add,axiom,
    ! [K_1: real,Xa_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1))))) ).

tff(fact_4741_less__eq__Suc__le__raw,axiom,
    ! [X_2: nat] : hAPP_n1699378549t_bool(ord_less_nat,X_2) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_2)) ).

tff(fact_4742_DERIV__sin__sin__mult,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))))) ).

tff(fact_4743_DERIV__sin__add,axiom,
    ! [K_1: real,Xa_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1)))) ).

tff(fact_4744_bdd__nat__set__le__finite,axiom,
    ! [X_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_1)))) ).

tff(fact_4745_card__Collect__le__nat,axiom,
    ! [N_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),N_1))) = hAPP_nat_nat(suc,N_1) ).

tff(fact_4746_card__bdd__nat__set__le,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_1))) = hAPP_nat_nat(suc,X_1) ).

tff(fact_4747_nat__number__of__Bit0,axiom,
    ! [W: int] : hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W)) ).

tff(fact_4748_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J,axiom,
    ! [P_1: fun_int_bool] : hBOOL(hAPP_f448129468l_bool(nat_nat_set,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)))) ).

tff(fact_4749_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J,axiom,
    ! [P_1: fun_nat_bool] : collect_nat(P_1) = image_int_nat(nat_1,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P_1),nat_1)))) ).

tff(fact_4750_Bseq__realpow,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))
       => hBOOL(bseq_real(hAPP_r474017924t_real(power_power_real,X_1))) ) ) ).

tff(fact_4751_bdd__int__set__le__finite,axiom,
    ! [N_1: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1))))) ).

tff(fact_4752_diff__nat__eq__if,axiom,
    ! [Z_2: int,Z_3: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,Z_3))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,Z_2)),hAPP_int_nat(nat_1,Z_3)) = hAPP_int_nat(nat_1,Z_2) ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,Z_3))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,Z_2)),hAPP_int_nat(nat_1,Z_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))),zero_zero_nat),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))) ) ) ) ).

tff(fact_4753_finite__divisors__int,axiom,
    ! [I_1: int] :
      ( ( I_1 != zero_zero_int )
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,dvd_dvd_int),I_1)))) ) ).

tff(fact_4754_finite__divisors__nat,axiom,
    ! [Ma: nat] :
      ( ( Ma != zero_zero_nat )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,dvd_dvd_nat),Ma)))) ) ).

tff(fact_4755_primes__infinite,axiom,
    ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(prime))) ).

tff(fact_4756_cnj_OCauchy,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(cauchy_complex(X_3))
     => hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))) ) ).

tff(fact_4757_Im_OCauchy,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(cauchy_complex(X_3))
     => hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))) ) ).

tff(fact_4758_Re_OCauchy,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(cauchy_complex(X_3))
     => hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))) ) ).

tff(fact_4759_Ints__real__of__nat,axiom,
    ! [N_1: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N_1)),ring_1_Ints_real)) ).

tff(fact_4760_Ints__real__of__int,axiom,
    ! [X_1: int] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,X_1)),ring_1_Ints_real)) ).

tff(fact_4761_DERIV__fun__cos,axiom,
    ! [G: fun_real_real,X_1: real,Ma: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(G,X_1),Ma))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),G),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,X_1)))),Ma))) ) ).

tff(fact_4762_DERIV__fun__sin,axiom,
    ! [G: fun_real_real,X_1: real,Ma: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(G,X_1),Ma))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,X_1))),Ma))) ) ).

tff(fact_4763_bdd__nat__set__l__finite,axiom,
    ! [X_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_1)))) ).

tff(fact_4764_card__Collect__less__nat,axiom,
    ! [N_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N_1))) = N_1 ).

tff(fact_4765_card__bdd__nat__set__l,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_1))) = X_1 ).

tff(fact_4766_finite__M__bounded__by__nat,axiom,
    ! [P_1: fun_nat_bool,I_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),P_1)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) ).

tff(fact_4767_zpower__number__of__even,axiom,
    ! [Z_1: int,W: int] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) ).

tff(fact_4768_card__less__Suc2,axiom,
    ! [I_1: nat,M_2: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2))
     => ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) ) ) ).

tff(fact_4769_card__less,axiom,
    ! [I_1: nat,M_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2))
     => ( hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) != zero_zero_nat ) ) ).

tff(fact_4770_card__less__Suc,axiom,
    ! [I_1: nat,M_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2))
     => ( hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) ) ) ).

tff(fact_4771_bdd__int__set__l__l__finite,axiom,
    ! [N_1: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1))))) ).

tff(fact_4772_noXRRset__def,axiom,
    ! [Ma: int,X_1: int] : hAPP_i1948725293t_bool(noXRRset(Ma),X_1) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),hAPP_i1948725293t_bool(norRRset,Ma)) ).

tff(fact_4773_DERIV__fun__pow,axiom,
    ! [N_1: nat,G: fun_real_real,X_1: real,Ma: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(G,X_1),Ma))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),G)),N_1),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)))),Ma))) ) ).

tff(fact_4774_card__bdd__int__set__le,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat_1,N_1)),one_one_nat) ) ) ).

tff(fact_4775_int__card__bdd__int__set__l__l,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1))))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int) ) ) ).

tff(fact_4776_card__bdd__int__set__l__l,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1))
     => ( hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat_1,N_1)),one_one_nat) ) ) ).

tff(fact_4777_DERIV__cos__cos__mult,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))))) ).

tff(fact_4778_Suc__eq__number__of,axiom,
    ! [N_1: nat,V: int] :
      ( ( hAPP_nat_nat(suc,N_1) = hAPP_int_nat(number_number_of_nat,V) )
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => ( hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N_1 ) ) ) ) ).

tff(fact_4779_eq__number__of__Suc,axiom,
    ! [V: int,N_1: nat] :
      ( ( hAPP_int_nat(number_number_of_nat,V) = hAPP_nat_nat(suc,N_1) )
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => ( hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N_1 ) ) ) ) ).

tff(fact_4780_nat__number__of__Bit1,axiom,
    ! [W: int] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = zero_zero_nat ) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) ) ) ) ).

tff(fact_4781_DERIV__pow,axiom,
    ! [N_1: nat,X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N_1),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))))) ).

tff(fact_4782_DERIV__cos__cos__mult2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4783_less__number__of__Suc,axiom,
    ! [V: int,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N_1)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N_1)) ) ) ).

tff(fact_4784_less__Suc__number__of,axiom,
    ! [N_1: nat,V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))) ) ) ) ).

tff(fact_4785_le__Suc__number__of,axiom,
    ! [N_1: nat,V: int] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))) ) ) ) ).

tff(fact_4786_le__number__of__Suc,axiom,
    ! [V: int,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N_1)))
    <=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N_1)) ) ) ).

tff(fact_4787_SR__pos,axiom,
    ! [X_3: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(standardRes(Ma),X_3)),collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Ma))))) ) ).

tff(fact_4788_RsetR__zmult__mono,axiom,
    ! [X_1: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
       => ( ( hAPP_int_int(legacy_zgcd(X_1),Ma) = one_one_int )
         => hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),A_1)),rsetR(Ma))) ) ) ) ).

tff(fact_4789_zpower__number__of__odd,axiom,
    ! [Z_1: int,W: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)))
       => ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = one_one_int ) ) ) ).

tff(fact_4790_DERIV__sin__sin__mult2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4791_DERIV__log,axiom,
    ! [B: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(deriv_real(log(B),X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(ln,B)),X_1)))) ) ).

tff(fact_4792_DERIV__cos__cos__mult3,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) ).

tff(fact_4793_DERIV__sin__circle__all__zero,axiom,
    ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real)) ).

tff(fact_4794_DERIV__sin__realpow2a,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4795_DERIV__sin__realpow2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))))) ).

tff(fact_4796_diff__nat__number__of,axiom,
    ! [V: int,V_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,V) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))),zero_zero_nat),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))) ) ) ) ).

tff(fact_4797_DERIV__cos__realpow2a,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))) ).

tff(fact_4798_DERIV__cos__realpow2b,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) ).

tff(fact_4799_DERIV__cos__realpow2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))))) ).

tff(fact_4800_lemma__DERIV__tan,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) != zero_zero_real )
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),sin)),cos),X_1),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).

tff(fact_4801_lemma__DERIV__sin__cos__add,axiom,
    ! [Y_2: real,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real)) ).

tff(fact_4802_DERIV__sin__circle__all,axiom,
    ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))))) ).

tff(fact_4803_lemma__DERIV__sin__cos__minus,axiom,
    ! [X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),uminus_uminus_real))),sin))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),uminus_uminus_real))),cos))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real)) ).

tff(fact_4804_norm__frac_Opsimps,axiom,
    ! [A: int,B: int] :
      ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B)))
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
         => ( norm_frac(A,B) = norm_frac(hAPP_int_int(uminus_uminus_int,A),hAPP_int_int(uminus_uminus_int,B)) ) )
        & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int))
         => ( ( ( ( A = zero_zero_int )
                | ( B = zero_zero_int ) )
             => ( norm_frac(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
            & ( ~ ( ( A = zero_zero_int )
                  | ( B = zero_zero_int ) )
             => ( norm_frac(A,B) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))) ) ) ) ) ) ) ).

tff(fact_4805_monoseq__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(monoseq_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) ) ).

tff(fact_4806_monoseq__realpow,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))
       => hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,X_1))) ) ) ).

tff(fact_4807_norm__frac_Osimps,axiom,
    ! [A_3: int,B_2: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
       => ( norm_frac(A_3,B_2) = norm_frac(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int))
       => ( ( ( ( A_3 = zero_zero_int )
              | ( B_2 = zero_zero_int ) )
           => ( norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
          & ( ~ ( ( A_3 = zero_zero_int )
                | ( B_2 = zero_zero_int ) )
           => ( norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))) ) ) ) ) ) ).

tff(fact_4808_arg__def,axiom,
    ! [Z_2: complex] : arg(Z_2) = hilbert_Eps_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),cos))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),sin))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,pi)))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))))) ).

tff(fact_4809_Union__SetS__setprod__prop1,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_5)) ) ) ) ) ).

tff(fact_4810_d22set__prod__zfact,axiom,
    ! [A: int] : big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,A)) = hAPP_int_int(zfact,A) ).

tff(fact_4811_Bnor__prod__zgcd,axiom,
    ! [A: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma))
     => ( hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),Ma) = one_one_int ) ) ).

tff(fact_4812_Bnor__prod__power,axiom,
    ! [A: int,Ma: int,X_1: int] :
      ( ( X_1 != zero_zero_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma))
       => ( big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),hAPP_i1948725293t_bool(bnorRset(A),Ma))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),Ma)))) ) ) ) ).

tff(fact_4813_SetS__setprod__prop,axiom,
    ! [X_1: fun_int_bool,A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A))
           => ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(A,P_5)))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,X_1)),A),P_5)) ) ) ) ) ) ).

tff(fact_4814_wset__zcong__prod__1,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(wset(A),P_5))),one_one_int),P_5)) ) ) ) ).

tff(fact_4815_Union__SetS__setprod__prop2,axiom,
    ! [A: int,P_5: int] :
      ( hBOOL(hAPP_int_bool(zprime,P_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5))
         => ( big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5))) = hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)) ) ) ) ) ).

tff(fact_4816_transfer__nat__int__sum__prod__closure_I2_J,axiom,
    ! [F: fun_int_int,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_2))) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A_1))) ) ) ).

tff(fact_4817_arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => ( hAPP_real_real(arctan,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))) ) ) ).

tff(fact_4818_DERIV__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))))),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),X_1),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ).

tff(fact_4819_lemma__sin__ext,axiom,
    ! [X_2: real] : hAPP_real_real(sin,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) ).

tff(fact_4820_sin__def,axiom,
    ! [X_1: real] : hAPP_real_real(sin,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) ).

tff(fact_4821_lemma__cos__ext,axiom,
    ! [X_2: real] : hAPP_real_real(cos,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) ).

tff(fact_4822_cos__def,axiom,
    ! [X_1: real] : hAPP_real_real(cos,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) ).

tff(fact_4823_lemma__sin__minus,axiom,
    ! [X_1: real] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1)) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4824_exp__first__two__terms,axiom,
    ! [X_1: real] : hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_4825_pi__series,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),one_one_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))) ).

tff(fact_4826_ln__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
       => ( hAPP_real_real(ln,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real))),suc))) ) ) ) ).

tff(fact_4827_summable__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))) ) ).

tff(fact_4828_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))
     => hBOOL(summable_real(F)) ) ).

tff(fact_4829_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N_3: nat] :
        ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2))) )
     => ( hBOOL(summable_real(G))
       => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) ) ) ).

tff(fact_4830_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,F))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))) ) ).

tff(fact_4831_summable__sin,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4832_summable__cos,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4833_rat__floor__code,axiom,
    ! [P_5: rat] : archim791455193or_rat(P_5) = hAPP_P1175774780nt_int(produc1298267108nt_int(div_div_int),quotient_of(P_5)) ).

tff(fact_4834_rat__uminus__code,axiom,
    ! [P_5: rat] : quotient_of(hAPP_rat_rat(uminus_uminus_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),quotient_of(P_5)) ).

tff(fact_4835_summable__exp,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1)))) ).

tff(fact_4836_rat__abs__code,axiom,
    ! [P_5: rat] : quotient_of(hAPP_rat_rat(abs_abs_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),abs_abs_int)),quotient_of(P_5)) ).

tff(fact_4837_divmod__int__rel__def,axiom,
    ! [A: int,B: int] : divmod_int_rel(A,B) = produc93441119t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool),hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj)),hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(fequal_int,A))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,B)))))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),B))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_bool_bool(fNot,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,B))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),zero_zero_int)))))) ).

tff(fact_4838_sum2sq__def,axiom,
    twoSqu2000912462sum2sq = produc1298267108nt_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,cOMBB_int_int_int),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,power_power_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,power_power_int),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4839_rat__inverse__code,axiom,
    ! [P_5: rat] : quotient_of(hAPP_rat_rat(inverse_inverse_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,cOMBB_1389251822nt_int),hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)))),hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,cOMBC_1964283556nt_int),hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int)),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,times_times_int),sgn_sgn_int)))),abs_abs_int))),quotient_of(P_5)) ).

tff(fact_4840_exp__tail__after__first__two__terms__summable,axiom,
    ! [X_1: real] : hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).

tff(fact_4841_adjust__def,axiom,
    ! [B: int] : adjust(B) = produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_4842_DERIV__power__series_H,axiom,
    ! [X0: real,F: fun_nat_real,R_4: real] :
      ( ! [X_2: real] :
          ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4)))
         => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X_2)))) )
     => ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4)))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_4))
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),suc))),X0),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X0))))) ) ) ) ).

tff(fact_4843_Nitpick_OFrac__def,axiom,
    frac = produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)))),hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,cOMBC_int_int_bool),hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,hAPP_f1399575567t_bool(cOMBB_118231410ol_int,fequal_int)),int_gcd))),one_one_int))) ).

tff(fact_4844_cnj_Osuminf,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => ( hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)) ) ) ).

tff(fact_4845_cnj_Osummable,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))) ) ).

tff(fact_4846_Re_Osummable,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))) ) ).

tff(fact_4847_Im_Osummable,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))) ) ).

tff(fact_4848_Re_Osuminf,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => ( hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)) ) ) ).

tff(fact_4849_Im_Osuminf,axiom,
    ! [X_3: fun_nat_complex] :
      ( hBOOL(summable_complex(X_3))
     => ( hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_3)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)) ) ) ).

tff(fact_4850_ratrel__def,axiom,
    ratrel = collec50511176nt_int(produc973645403t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,cOMBB_1344650379nt_int),hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,fNot),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),product_snd_int_int)),zero_zero_int))))),hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,fNot),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),product_snd_int_int)),zero_zero_int))))),hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,cOMBS_480885903t_bool),hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int)),hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,cOMBB_47643171nt_int),hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,times_times_int),product_fst_int_int))),product_snd_int_int)))),hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,hAPP_f578362681nt_int(cOMBC_1417754532nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,times_times_int),product_fst_int_int))),product_snd_int_int)))))) ).

tff(fact_4851_divmod__nat__if,axiom,
    ! [Ma: nat,N_1: nat] :
      ( ( ( ( N_1 = zero_zero_nat )
          | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) )
       => ( divmod_nat(Ma,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),Ma) ) )
      & ( ~ ( ( N_1 = zero_zero_nat )
            | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) )
       => ( divmod_nat(Ma,N_1) = hAPP_P1586233937at_nat(produc1391996073at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,product_Pair_nat_nat),suc)),divmod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N_1),N_1)) ) ) ) ).

tff(fact_4852_Bolzano__bisect_Osimps_I2_J,axiom,
    ! [P_1: fun_Pr1127657839l_bool,A: real,B: real,N_1: nat] : hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_1)) = hAPP_P1860904029l_real(produc595218619l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,cOMBS_902912708l_real),hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,cOMBS_794971624l_real),hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,hAPP_f1076489843l_real(cOMBB_415306361l_real,if_Pro313124157l_real)),hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,hAPP_f940061907l_bool(cOMBB_1795223523l_real,P_1)),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,cOMBS_511527878l_real),hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,hAPP_f2114902373l_real(cOMBB_237479429l_real,produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),cOMBI_real)))),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1)) ).

tff(fact_4853_suminf__exist__split,axiom,
    ! [A: fun_nat_real,R_2: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2))
     => ( hBOOL(summable_real(A))
       => ? [N_3: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,A),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_2))))),R_2)) ) ) ) ).

tff(fact_4854_divmod__int__pdivmod,axiom,
    ! [L: int,K_1: int] :
      ( ( ( K_1 = zero_zero_int )
       => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ) )
      & ( ( K_1 != zero_zero_int )
       => ( ( ( L = zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) ) )
          & ( ( L != zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fdisj(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)),hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,L),zero_zero_int)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) ) ) ) ) ) ).

tff(fact_4855_negateSnd__def,axiom,
    negateSnd = produc713050258nt_int(uminus_uminus_int) ).

tff(fact_4856_divmod__int__code,axiom,
    ! [L: int,K_1: int] :
      ( ( ( K_1 = zero_zero_int )
       => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) ) )
      & ( ( K_1 != zero_zero_int )
       => ( ( ( L = zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) ) )
          & ( ( L != zero_zero_int )
           => ( divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K_1)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) ) ) ) ) ) ).

tff(fact_4857_summable__le2,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2)))
     => ( hBOOL(summable_real(G))
       => ( hBOOL(summable_real(F))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) ) ) ) ).

tff(fact_4858_suminf__gt__zero,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_2)))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4859_suminf__ge__zero,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4860_suminf__0__le,axiom,
    ! [F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
     => ( hBOOL(summable_real(F))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4861_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,cOMBB_bool_bool_int),hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) ).

tff(fact_4862_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int))) ).

tff(fact_4863_sin__paired,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))),hAPP_real_real(sin,X_1))) ).

tff(fact_4864_sin__converges,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_real_real(sin,X_1))) ).

tff(fact_4865_cos__converges,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_real_real(cos,X_1))) ).

tff(fact_4866_power__half__series,axiom,
    hBOOL(hAPP_real_bool(sums_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),suc)),one_one_real)) ).

tff(fact_4867_Ln_Oaux2,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).

tff(fact_4868_sums__if_H,axiom,
    ! [G: fun_nat_real,X_1: real] :
      ( hBOOL(hAPP_real_bool(sums_real(G),X_1))
     => hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),X_1)) ) ).

tff(fact_4869_sums__if,axiom,
    ! [F: fun_nat_real,Y_2: real,G: fun_nat_real,X_1: real] :
      ( hBOOL(hAPP_real_bool(sums_real(G),X_1))
     => ( hBOOL(hAPP_real_bool(sums_real(F),Y_2))
       => hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2))) ) ) ).

tff(fact_4870_cos__paired,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(cos,X_1))) ).

tff(fact_4871_cnj_Osums,axiom,
    ! [X_3: fun_nat_complex,A: complex] :
      ( hBOOL(hAPP_complex_bool(sums_complex(X_3),A))
     => hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)),hAPP_complex_complex(cnj,A))) ) ).

tff(fact_4872_Re_Osums,axiom,
    ! [X_3: fun_nat_complex,A: complex] :
      ( hBOOL(hAPP_complex_bool(sums_complex(X_3),A))
     => hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)),hAPP_complex_real(re,A))) ) ).

tff(fact_4873_Im_Osums,axiom,
    ! [X_3: fun_nat_complex,A: complex] :
      ( hBOOL(hAPP_complex_bool(sums_complex(X_3),A))
     => hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)),hAPP_complex_real(im,A))) ) ).

tff(fact_4874_sin__fdiffs2,axiom,
    ! [N_1: nat] : hAPP_nat_real(diffs_real(sin_coeff),N_1) = hAPP_nat_real(cos_coeff,N_1) ).

tff(fact_4875_sin__fdiffs,axiom,
    diffs_real(sin_coeff) = cos_coeff ).

tff(fact_4876_exp__fdiffs,axiom,
    ! [X_2: nat] : hAPP_nat_real(diffs_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),X_2) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2))) ).

tff(fact_4877_less__eq__nat_Osimps_I2_J,axiom,
    ! [Ma: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N_1))
    <=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),N_1)) ) ).

tff(fact_4878_cos__fdiffs2,axiom,
    ! [N_1: nat] : hAPP_nat_real(diffs_real(cos_coeff),N_1) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,N_1)) ).

tff(fact_4879_cos__fdiffs,axiom,
    ! [X_2: nat] : hAPP_nat_real(diffs_real(cos_coeff),X_2) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X_2)) ).

tff(fact_4880_diff__Suc,axiom,
    ! [Ma: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N_1)) ).

tff(fact_4881_summable__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(G,N_2)))
     => ( hBOOL(summable_real(F))
       => ( hBOOL(summable_real(G))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) ) ) ) ).

tff(fact_4882_arctan__def,axiom,
    ! [Y_2: real] : hAPP_real_real(arctan,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),tan)),Y_2)))) ).

tff(fact_4883_arcsin__def,axiom,
    ! [Y_2: real] : hAPP_real_real(arcsin,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),sin)),Y_2)))) ).

tff(fact_4884_ln__def,axiom,
    ! [X_1: real] : hAPP_real_real(ln,X_1) = the_real(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),exp_real)),X_1)) ).

tff(fact_4885_root__def,axiom,
    ! [N_1: nat,X_1: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N_1))),X_1))) ) )
      & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
           => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = hAPP_real_real(uminus_uminus_real,the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N_1))),hAPP_real_real(uminus_uminus_real,X_1))))) ) )
          & ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
           => ( hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) = zero_zero_real ) ) ) ) ) ).

tff(fact_4886_arccos__def,axiom,
    ! [Y_2: real] : hAPP_real_real(arccos,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),Y_2)))) ).

tff(fact_4887_pi__def,axiom,
    pi = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),zero_zero_real))))) ).

tff(fact_4888_pi__half,axiom,
    hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),zero_zero_real)))) ).

tff(fact_4889_divmod__nat__def,axiom,
    ! [Ma: nat,N_1: nat] : divmod_nat(Ma,N_1) = the_Pr588456374at_nat(divmod_nat_rel(Ma,N_1)) ).

tff(fact_4890_floor__real__def,axiom,
    ! [X_1: real] : archim1246769320r_real(X_1) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,ord_less_eq_real),ring_1_of_int_real)),X_1))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,X_1)),hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,ring_1_of_int_real),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) ).

tff(fact_4891_floor__rat__def,axiom,
    ! [X_1: rat] : archim791455193or_rat(X_1) = the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,ord_less_eq_rat),ring_1_of_int_rat)),X_1))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,X_1)),hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,ring_1_of_int_rat),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) ).

tff(fact_4892_RRset2norRR__def,axiom,
    ! [A: int,A_1: fun_int_bool,Ma: int] :
      ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
          & hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
          & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) )
       => ( hAPP_int_int(rRset2norRR(A_1,Ma),A) = hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma)))) ) )
      & ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
            & hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) )
       => ( hAPP_int_int(rRset2norRR(A_1,Ma),A) = zero_zero_int ) ) ) ).

tff(fact_4893_aux__some,axiom,
    ! [A: int,A_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
     => ( hBOOL(hAPP_int_bool(is_RRset(A_1),Ma))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),Ma))
            & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ) ).

tff(fact_4894_Maclaurin__sin__bound,axiom,
    ! [X_1: real,N_1: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,X_1)),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,X_1)),N_1)))) ).

tff(fact_4895_finite__less__ub,axiom,
    ! [U: nat,F: fun_nat_nat] :
      ( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))) ) ).

tff(fact_4896_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U))) ).

tff(fact_4897_sumr__diff__mult__const,axiom,
    ! [F: fun_nat_real,N_1: nat,R_2: real] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),R_2)) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),F)),R_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)) ).

tff(fact_4898_sumr__offset2,axiom,
    ! [K_1: nat,N_1: nat,F_2: fun_nat_real] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) ).

tff(fact_4899_sumr__offset4,axiom,
    ! [K_1: nat,N_2: nat,F_2: fun_nat_real] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_1))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) ).

tff(fact_4900_sumr__one__lb__realpow__zero,axiom,
    ! [F: fun_nat_real,N_1: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,zero_zero_nat)),N_1)) = zero_zero_real ).

tff(fact_4901_ex__nat__less__eq,axiom,
    ! [P_1: fun_nat_bool,N_1: nat] :
      ( ? [M_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
          & hBOOL(hAPP_nat_bool(P_1,M_1)) )
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)))
          & hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_4902_all__nat__less__eq,axiom,
    ! [P_1: fun_nat_bool,N_1: nat] :
      ( ! [M_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
         => hBOOL(hAPP_nat_bool(P_1,M_1)) )
    <=> ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)))
         => hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_4903_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),L) ).

tff(fact_4904_sumr__geometric,axiom,
    ! [N_1: nat,X_1: real] :
      ( ( X_1 != one_one_real )
     => ( hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real)) ) ) ).

tff(fact_4905_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,L)),U) = hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U) ).

tff(fact_4906_image__add__atLeastLessThan,axiom,
    ! [K_1: nat,I_1: nat,J: nat] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).

tff(fact_4907_image__Suc__atLeastLessThan,axiom,
    ! [I_1: nat,J: nat] : image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) ).

tff(fact_4908_Maclaurin__zero,axiom,
    ! [Diff: fun_na182379510l_real,N_1: nat,X_1: real] :
      ( ( X_1 = zero_zero_real )
     => ( ( N_1 != zero_zero_nat )
       => ( hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)) = hAPP_real_real(hAPP_n546711566l_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

tff(fact_4909_subset__card__intvl__is__intvl,axiom,
    ! [A_1: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1)))))
     => ( A_1 = hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1))) ) ) ).

tff(fact_4910_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N_1: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),F)),G)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))) ).

tff(fact_4911_sumr__minus__one__realpow__zero,axiom,
    ! [N_1: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),suc)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) = zero_zero_real ).

tff(fact_4912_lemma__STAR__cos2,axiom,
    ! [N_1: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,zero_zero_real)))),zero_zero_real)),hAPP_n1699378549t_bool(ord_at4362885an_nat(one_one_nat),N_1)) = zero_zero_real ).

tff(fact_4913_sumr__cos__zero__one,axiom,
    ! [N_1: nat] : hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(suc,N_1))) = one_one_real ).

tff(fact_4914_Maclaurin__sin__expansion2,axiom,
    ! [N_1: nat,X_1: real] :
    ? [T: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
      & ( hAPP_real_real(sin,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N_1))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ).

tff(fact_4915_Maclaurin__cos__expansion,axiom,
    ! [N_1: nat,X_1: real] :
    ? [T: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
      & ( hAPP_real_real(cos,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N_1))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ).

tff(fact_4916_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U))) ).

tff(fact_4917_Sup__atLeastLessThan,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1))
     => ( comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Y_2),X_1)) = X_1 ) ) ).

tff(fact_4918_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U))) ).

tff(fact_4919_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)) = hAPP_int_nat(nat_1,U) ).

tff(fact_4920_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)) ).

tff(fact_4921_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : hAPP_i1948725293t_bool(ord_at641636577an_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U) ).

tff(fact_4922_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),L),hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L))) = hAPP_i1948725293t_bool(ord_at641636577an_int(L),U) ).

tff(fact_4923_Maclaurin__sin__expansion,axiom,
    ! [X_1: real,N_1: nat] :
    ? [T: real] : hAPP_real_real(sin,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N_1))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ).

tff(fact_4924_Maclaurin__lemma2,axiom,
    ! [B_1: real,K_1: nat,Diff: fun_na182379510l_real,H: real,N_1: nat] :
      ( ! [M_1: nat,T: real] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
     => ( ( N_1 = hAPP_nat_nat(suc,K_1) )
       => ! [M_1: nat,T: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_n546711566l_real(Diff,M_1))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,M_1))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),power_power_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B_1)),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1))))))),T),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,T))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,T),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,M_1)))))))))) ) ) ) ).

tff(fact_4925_Maclaurin__exp__lt,axiom,
    ! [N_1: nat,X_1: real] :
      ( ( X_1 != zero_zero_real )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ? [T: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))
            & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
            & ( hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ) ) ).

tff(fact_4926_Maclaurin__exp__le,axiom,
    ! [N_1: nat,X_1: real] :
    ? [T: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
      & ( hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ).

tff(fact_4927_sumr__pos__lt__pair,axiom,
    ! [K_1: nat,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [D_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)))),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)),one_one_nat))))))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4928_Maclaurin__lemma,axiom,
    ! [F: fun_real_real,J: fun_nat_real,N_1: nat,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
     => ? [B_6: real] : hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),J)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_6),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) ) ).

tff(fact_4929_series__pos__less,axiom,
    ! [N_1: nat,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [M_1: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,M_1))) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4930_series__pos__le,axiom,
    ! [N_1: nat,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [M_1: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,M_1))) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).

tff(fact_4931_Maclaurin__all__lt,axiom,
    ! [X_1: real,N_1: nat,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( ( X_1 != zero_zero_real )
         => ( ! [M_1: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))
           => ? [T: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
                & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ) ) ) ) ).

tff(fact_4932_Maclaurin__all__lt__objl,axiom,
    ! [N_1: nat,X_1: real,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_1: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))
        & ( X_1 != zero_zero_real )
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) )
     => ? [T: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
          & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ) ).

tff(fact_4933_Maclaurin__bi__le,axiom,
    ! [X_1: real,N_1: nat,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
     => ( ! [M_1: nat,T: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
       => ? [T: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
            & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ) ) ).

tff(fact_4934_Maclaurin__objl,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N_1: nat,H: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
        & ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_1: nat,T: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) ) )
     => ? [T: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H))
          & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) ) ) ) ).

tff(fact_4935_Maclaurin2__objl,axiom,
    ! [N_1: nat,Diff: fun_na182379510l_real,F: fun_real_real,H: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
        & ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_1: nat,T: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) ) )
     => ? [T: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H))
          & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) ) ) ) ).

tff(fact_4936_Maclaurin__minus,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N_1: nat,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
         => ( ! [M_1: nat,T: real] :
                ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) )
               => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
           => ? [T: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real))
                & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) ) ) ) ) ) ) ).

tff(fact_4937_Maclaurin2,axiom,
    ! [N_1: nat,Diff: fun_na182379510l_real,F: fun_real_real,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
     => ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
       => ( ! [M_1: nat,T: real] :
              ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
             => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
         => ? [T: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H))
              & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) ) ) ) ) ) ).

tff(fact_4938_Maclaurin__minus__objl,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N_1: nat,H: real] :
      ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
        & ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_1: nat,T: real] :
            ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) )
           => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) ) )
     => ? [T: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real))
          & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) ) ) ) ).

tff(fact_4939_Maclaurin,axiom,
    ! [Diff: fun_na182379510l_real,F: fun_real_real,N_1: nat,H: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
       => ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
         => ( ! [M_1: nat,T: real] :
                ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T))
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) )
               => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))) )
           => ? [T: real] :
                ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H))
                & ( hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) ) ) ) ) ) ) ).

tff(fact_4940_Maclaurin__all__le__objl,axiom,
    ! [N_1: nat,X_1: real,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
        & ! [M_1: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2))) )
     => ? [T: real] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
          & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ) ).

tff(fact_4941_Maclaurin__all__le,axiom,
    ! [N_1: nat,X_1: real,Diff: fun_na182379510l_real,F: fun_real_real] :
      ( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F )
     => ( ! [M_1: nat,X_2: real] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))
       => ? [T: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))
            & ( hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) ) ) ) ) ).

tff(fact_4942_real__setsum__nat__ivl__bounded,axiom,
    ! [F: fun_nat_real,K_3: real,N_1: nat] :
      ( ! [P_4: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),N_1))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,P_4)),K_3)) )
     => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),K_3))) ) ).

tff(fact_4943_DERIV__sumr,axiom,
    ! [F: fun_na182379510l_real,X_1: real,F_1: fun_na182379510l_real,N_1: nat,Ma: nat] :
      ( ! [R_1: nat] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),R_1))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1))) )
         => hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),X_1),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),X_1))) )
     => hBOOL(hAPP_real_bool(deriv_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f42270917t_real(cOMBC_nat_real_real,F))),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N_1)),X_1),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),X_1)),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N_1)))) ) ).

tff(fact_4944_suminf__le,axiom,
    ! [X_1: real,F: fun_nat_real] :
      ( hBOOL(summable_real(F))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_1))
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),X_1)) ) ) ).

tff(fact_4945_transfer__nat__int__sum__prod__closure_I1_J,axiom,
    ! [F: fun_int_int,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_2))) )
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A_1))) ) ) ).

tff(fact_4946_pos__summable,axiom,
    ! [X_1: real,F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_1))
       => hBOOL(summable_real(F)) ) ) ).

tff(fact_4947_arith__series__int,axiom,
    ! [A: int,D: int,N_1: nat] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,hAPP_int_fun_int_int(plus_plus_int,A)),hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,times_times_int),semiri1621563631at_int)),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))),D)))) ).

tff(fact_4948_BseqI2,axiom,
    ! [K_3: real,K_1: real,F: fun_nat_real] :
      ( ! [N_2: nat] :
          ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K_1),hAPP_nat_real(F,N_2)))
          & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),K_3)) )
     => hBOOL(bseq_real(F)) ) ).

tff(fact_4949_finite__lessThan,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ).

tff(fact_4950_card__lessThan,axiom,
    ! [U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U ).

tff(fact_4951_atLeast0LessThan,axiom,
    ! [N_1: nat] : hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1) = hAPP_n1699378549t_bool(ord_lessThan_nat,N_1) ).

tff(fact_4952_finite__nat__iff__bounded,axiom,
    ! [S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ? [K: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ) ).

tff(fact_4953_arith__series__nat,axiom,
    ! [A: nat,D: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(plus_plus_nat,A)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),D)))) ).

tff(fact_4954_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U))
     => ( hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_int_nat(nat_1,U))) ) ) ).

tff(fact_4955_finite__nat__bounded,axiom,
    ! [S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
     => ? [K: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ) ).

tff(fact_4956_sum__diff__distrib,axiom,
    ! [N_1: nat,Q_3: fun_nat_nat,P_1: fun_nat_nat] :
      ( ! [X_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_3,X_2)),hAPP_nat_nat(P_1,X_2)))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,P_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_3),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),P_1)),Q_3)),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1)) ) ) ).

tff(fact_4957_Sup__lessThan,axiom,
    ! [X_1: real] : comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,X_1)) = X_1 ).

tff(fact_4958_transfer__nat__int__set__cong,axiom,
    ! [P_2: fun_int_bool,P_1: fun_int_bool] :
      ( ! [X_2: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => ( hBOOL(hAPP_int_bool(P_1,X_2))
          <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
     => ( collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_2)) ) ) ).

tff(fact_4959_normalize__def,axiom,
    ! [P_3: product_prod_int_int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))
       => ( normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))) ) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))
       => ( ( ( hAPP_P1175774780nt_int(product_snd_int_int,P_3) = zero_zero_int )
           => ( normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
          & ( ( hAPP_P1175774780nt_int(product_snd_int_int,P_3) != zero_zero_int )
           => ( normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))) ) ) ) ) ) ).

tff(fact_4960_gcd__dvd1__int,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),X)) ).

tff(fact_4961_gcd__dvd2__int,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),Y)) ).

tff(fact_4962_coprime__exp2__int,axiom,
    ! [N: nat,M: nat,A_3: int,B_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),M)) = one_one_int ) ) ).

tff(fact_4963_invertible__coprime__int,axiom,
    ! [X: int,Y: int,M: int] :
      ( ( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),M) = one_one_int ) ) ).

tff(fact_4964_gcd__non__0__int,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) ) ) ).

tff(fact_4965_gcd__coprime__int,axiom,
    ! [B_5: int,A_5: int,A_3: int,B_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int )
     => ( ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) )
       => ( ( B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) )
         => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_5),B_5) = one_one_int ) ) ) ) ).

tff(fact_4966_transfer__int__nat__gcd_I1_J,axiom,
    ! [X: nat,Y: nat] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).

tff(fact_4967_gcd__add__mult__int,axiom,
    ! [M: int,K_2: int,N: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ).

tff(fact_4968_gcd__add1__int,axiom,
    ! [M: int,N: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ).

tff(fact_4969_gcd__add2__int,axiom,
    ! [M: int,N: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ).

tff(fact_4970_coprime__plus__one__int,axiom,
    ! [N: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),N) = one_one_int ).

tff(fact_4971_coprime__minus__one__int,axiom,
    ! [N: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int)),N) = one_one_int ).

tff(fact_4972_coprime__divisors__nat,axiom,
    ! [E_1: int,B_2: int,D_3: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),A_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E_1),B_2))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int )
         => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),E_1) = one_one_int ) ) ) ) ).

tff(fact_4973_coprime__exp__int,axiom,
    ! [N: nat,D_3: int,A_3: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)) = one_one_int ) ) ).

tff(fact_4974_gcd__1__int,axiom,
    ! [M: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

tff(fact_4975_gcd__zero__int,axiom,
    ! [Ma: int,N_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N_1) = zero_zero_int )
    <=> ( ( Ma = zero_zero_int )
        & ( N_1 = zero_zero_int ) ) ) ).

tff(fact_4976_gcd__exp__int,axiom,
    ! [A_3: int,N: nat,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),N) ).

tff(fact_4977_gcd__commute__int,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),A_3) ).

tff(fact_4978_gcd__int_Oleft__commute,axiom,
    ! [B_2: int,A_3: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) ).

tff(fact_4979_gcd__assoc__int,axiom,
    ! [A_3: int,B_2: int,C_1: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) ).

tff(fact_4980_dvd__gcd__D2__int,axiom,
    ! [I: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N)) ) ).

tff(fact_4981_dvd__gcd__D1__int,axiom,
    ! [I: int,M: int,N: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),M)) ) ).

tff(fact_4982_gcd__greatest__int,axiom,
    ! [N: int,K_2: int,M: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N))) ) ) ).

tff(fact_4983_gcd__greatest__iff__int,axiom,
    ! [K_1: int,Ma: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N_1)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) ) ) ).

tff(fact_4984_gcd__red__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) ).

tff(fact_4985_abs__gcd__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4986_gcd__idem__int,axiom,
    ! [X: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),X) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_4987_gcd__abs__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) ).

tff(fact_4988_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(abs_abs_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4989_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4990_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4991_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(uminus_uminus_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).

tff(fact_4992_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),Y))
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,X) ) ) ).

tff(fact_4993_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X))
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,Y) ) ) ).

tff(fact_4994_gcd__mult__distrib__int,axiom,
    ! [K_2: int,M: int,N: int] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) ).

tff(fact_4995_gcd__pos__int,axiom,
    ! [Ma: int,N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N_1)))
    <=> ( ( Ma != zero_zero_int )
        | ( N_1 != zero_zero_int ) ) ) ).

tff(fact_4996_coprime__rmult__int,axiom,
    ! [D_3: int,A_3: int,B_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int ) ) ).

tff(fact_4997_coprime__lmult__int,axiom,
    ! [D_3: int,A_3: int,B_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int ) ) ).

tff(fact_4998_coprime__mult__int,axiom,
    ! [B_2: int,D_3: int,A_3: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int )
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int )
       => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int ) ) ) ).

tff(fact_4999_gcd__mult__cancel__int,axiom,
    ! [M: int,K_2: int,N: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) = one_one_int )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ) ) ).

tff(fact_5000_coprime__mul__eq__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)) = one_one_int )
    <=> ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),A) = one_one_int )
        & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),B) = one_one_int ) ) ) ).

tff(fact_5001_gcd__0__left__int,axiom,
    ! [X: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_5002_gcd__0__int,axiom,
    ! [X: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = hAPP_int_int(abs_abs_int,X) ).

tff(fact_5003_Fract__coprime,axiom,
    ! [A_3: int,B_2: int] : hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ).

tff(fact_5004_div__gcd__coprime__int,axiom,
    ! [B_2: int,A_3: int] :
      ( ( ( A_3 != zero_zero_int )
        | ( B_2 != zero_zero_int ) )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))) = one_one_int ) ) ).

tff(fact_5005_coprime__dvd__mult__int,axiom,
    ! [M: int,K_2: int,N: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).

tff(fact_5006_coprime__dvd__mult__iff__int,axiom,
    ! [Ma: int,K_1: int,N_1: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),N_1) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1)))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) ) ) ).

tff(fact_5007_divides__mult__int,axiom,
    ! [N: int,M: int,R_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),R_3))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),R_3))
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) = one_one_int )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)),R_3)) ) ) ) ).

tff(fact_5008_gcd__cases__int,axiom,
    ! [P_1: fun_int_bool,Y_2: int,X_1: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
         => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_2))) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int))
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(uminus_uminus_int,Y_2)))) ) )
       => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),zero_zero_int))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2))
             => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),Y_2))) ) )
         => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),zero_zero_int))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),hAPP_int_int(uminus_uminus_int,Y_2)))) ) )
           => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_2))) ) ) ) ) ).

tff(fact_5009_gcd__unique__int,axiom,
    ! [B: int,A: int,D: int] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),A))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),B))
        & ! [E: int] :
            ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),A))
              & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),B)) )
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D)) ) )
    <=> ( D = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

tff(fact_5010_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) ).

tff(fact_5011_transfer__nat__int__gcd__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ).

tff(fact_5012_gcd__le2__int,axiom,
    ! [A_3: int,B_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),B_2)) ) ).

tff(fact_5013_gcd__le1__int,axiom,
    ! [B_2: int,A_3: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),A_3)) ) ).

tff(fact_5014_coprime__crossproduct__int,axiom,
    ! [B: int,C: int,A: int,D: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),D) = one_one_int )
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B),C) = one_one_int )
       => ( ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A)),hAPP_int_int(abs_abs_int,C)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,B)),hAPP_int_int(abs_abs_int,D)) )
        <=> ( ( hAPP_int_int(abs_abs_int,A) = hAPP_int_int(abs_abs_int,B) )
            & ( hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D) ) ) ) ) ) ).

tff(fact_5015_gcd__code__int,axiom,
    ! [K_1: int,L: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) ).

tff(fact_5016_coprime__common__divisor__int,axiom,
    ! [X: int,A_3: int,B_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),A_3))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),B_2))
         => ( hAPP_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).

tff(fact_5017_quotient__of__coprime,axiom,
    ! [R_3: rat,P_3: int,Q_2: int] :
      ( ( quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int ) ) ).

tff(fact_5018_normalize__coprime,axiom,
    ! [R_3: product_prod_int_int,P_3: int,Q_2: int] :
      ( ( normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) )
     => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int ) ) ).

tff(fact_5019_coprime__int,axiom,
    ! [A: int,B: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) = one_one_int )
    <=> ! [D_2: int] :
          ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_2))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),A))
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),B)) )
        <=> ( D_2 = one_one_int ) ) ) ).

tff(fact_5020_transfer__nat__int__gcd_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat_1,X)),hAPP_int_nat(nat_1,Y)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ) ).

tff(fact_5021_gcd__int__def,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat_1,hAPP_int_int(abs_abs_int,Y)))) ).

tff(fact_5022_normalize__stable,axiom,
    ! [P_3: int,Q_2: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2))
     => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int )
       => ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) ) ) ) ).

tff(fact_5023_quotient__of__def,axiom,
    ! [X_1: rat] : quotient_of(X_1) = the_Pr2103884470nt_int(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,hAPP_r2115560837t_bool(fequal_rat,X_1)),hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,fract),product_fst_int_int)),product_snd_int_int)))),hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)),product_snd_int_int))),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,gcd_gcd_int),product_fst_int_int)),product_snd_int_int))),one_one_int)))) ).

tff(fact_5024_quotient__of__unique,axiom,
    ! [R_3: rat] :
    ? [X_2: product_prod_int_int] :
      ( ( R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) )
      & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X_2)))
      & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) = one_one_int )
      & ! [Y_1: product_prod_int_int] :
          ( ( ( R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) )
            & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)))
            & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) = one_one_int ) )
         => ( Y_1 = X_2 ) ) ) ).

tff(fact_5025_Rat__cases__nonzero,axiom,
    ! [Q_2: rat] :
      ( ! [A_2: int,B_4: int] :
          ( ( Q_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
           => ( ( A_2 != zero_zero_int )
             => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int ) ) ) )
     => ( Q_2 = zero_zero_rat ) ) ).

tff(fact_5026_Rat__cases,axiom,
    ! [Q_2: rat] :
      ~ ! [A_2: int,B_4: int] :
          ( ( Q_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
           => ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int ) ) ) ).

tff(fact_5027_Rat__induct,axiom,
    ! [Q_1: rat,P_1: fun_rat_bool] :
      ( ! [A_2: int,B_4: int] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4))
         => ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) = one_one_int )
           => hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))) ) )
     => hBOOL(hAPP_rat_bool(P_1,Q_1)) ) ).

tff(fact_5028_card__setsum__aux,axiom,
    ! [N_1: nat,S: fun_fu819804446l_bool] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S))
     => ( ! [X_2: fun_int_bool] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),S))
           => hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)) )
       => ( ! [X_2: fun_int_bool] :
              ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),S))
             => ( hAPP_f957591787ol_nat(finite_card_int,X_2) = N_1 ) )
         => ( hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,finite_card_int),S) = hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,cOMBK_1643584600t_bool(N_1)),S) ) ) ) ) ).

tff(fact_5029_zOdd__def,axiom,
    zOdd = collect_int(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int),hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,cOMBB_int_bool_int),fequal_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))) ).

tff(fact_5030_Rats__eq__int__div__int,axiom,
    field_1210416355s_real = collect_real(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,fEx_int),hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int)),hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,cOMBC_1683390479t_bool),hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool)),hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj))),hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,cOMBB_95956018ol_int),hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,cOMBB_real_bool_int),fequal_real))),hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,cOMBB_real_real_int),hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,inverse_divide_real),real_int))),real_int)))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fNot),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))))) ).

tff(fact_5031_zEven__def,axiom,
    zEven = collect_int(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,fEx_int),hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,cOMBB_int_bool_int),fequal_int)),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_5032_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(A))
       => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ).

tff(fact_5033_LIMSEQ__realpow__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,X_1),zero_zero_real,sequentially)) ) ) ).

tff(fact_5034_LIMSEQ__le,axiom,
    ! [Y_3: fun_nat_real,Y_2: real,X_3: fun_nat_real,X_1: real] :
      ( hBOOL(tendsto_nat_real(X_3,X_1,sequentially))
     => ( hBOOL(tendsto_nat_real(Y_3,Y_2,sequentially))
       => ( ? [N_3: nat] :
            ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
             => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),hAPP_nat_real(Y_3,N_2))) )
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) ) ) ) ).

tff(fact_5035_LIMSEQ__le__const2,axiom,
    ! [A: real,X_3: fun_nat_real,X_1: real] :
      ( hBOOL(tendsto_nat_real(X_3,X_1,sequentially))
     => ( ? [N_3: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),A)) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),A)) ) ) ).

tff(fact_5036_LIMSEQ__le__const,axiom,
    ! [A: real,X_3: fun_nat_real,X_1: real] :
      ( hBOOL(tendsto_nat_real(X_3,X_1,sequentially))
     => ( ? [N_3: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),hAPP_nat_real(X_3,N_2))) )
       => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_1)) ) ) ).

tff(fact_5037_monoseq__le,axiom,
    ! [X_1: real,A: fun_nat_real] :
      ( hBOOL(monoseq_real(A))
     => ( hBOOL(tendsto_nat_real(A,X_1,sequentially))
       => ( ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),X_1))
            & ! [M_1: nat,N_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,M_1)),hAPP_nat_real(A,N_2))) ) )
          | ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(A,N_2)))
            & ! [M_1: nat,N_2: nat] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2))
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),hAPP_nat_real(A,M_1))) ) ) ) ) ) ).

tff(fact_5038_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,X_1)),zero_zero_real,sequentially)) ) ).

tff(fact_5039_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R_2: real] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))),R_2,sequentially)) ).

tff(fact_5040_LIMSEQ__imp__rabs,axiom,
    ! [F: fun_nat_real,L: real] :
      ( hBOOL(tendsto_nat_real(F,L,sequentially))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),hAPP_real_real(abs_abs_real,L),sequentially)) ) ).

tff(fact_5041_LIMSEQ__rabs__zero,axiom,
    ! [F: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially))
    <=> hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)) ) ).

tff(fact_5042_LIMSEQ__inverse__real__of__nat,axiom,
    hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)),zero_zero_real,sequentially)) ).

tff(fact_5043_LIMSEQ__divide__realpow__zero,axiom,
    ! [A: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,A)),hAPP_r474017924t_real(power_power_real,X_1)),zero_zero_real,sequentially)) ) ).

tff(fact_5044_LIMSEQ__rabs__realpow__zero,axiom,
    ! [C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real))
     => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,C)),zero_zero_real,sequentially)) ) ).

tff(fact_5045_LIMSEQ__rabs__realpow__zero2,axiom,
    ! [C: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real))
     => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,C),zero_zero_real,sequentially)) ) ).

tff(fact_5046_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R_2: real] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),R_2,sequentially)) ).

tff(fact_5047_LIMSEQ__neg__powr,axiom,
    ! [Sa: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Sa))
     => hBOOL(tendsto_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,powr),real_nat)),hAPP_real_real(uminus_uminus_real,Sa)),zero_zero_real,sequentially)) ) ).

tff(fact_5048_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R_2: real] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,one_one_real)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))))),R_2,sequentially)) ).

tff(fact_5049_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(A))
       => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))) ) ) ).

tff(fact_5050_zeroseq__arctan__series,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real))
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),zero_zero_real,sequentially)) ) ).

tff(fact_5051_Rats__eq__int__div__nat,axiom,
    field_1210416355s_real = collect_real(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,fEx_int),hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,hAPP_f389116883t_bool(cOMBB_1593174431ol_int,fEx_nat)),hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,cOMBC_678732631t_bool),hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,cOMBS_nat_bool_bool)),hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,hAPP_f643658017l_bool(cOMBB_2046938128ol_int,hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj))),hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,cOMBB_1647910634ol_int),hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,cOMBB_real_bool_nat),fequal_real))),hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,cOMBB_real_real_nat),hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,inverse_divide_real),real_int))),real_nat)))))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),zero_zero_nat)))))) ).

tff(fact_5052_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(A))
       => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ).

tff(fact_5053_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
         => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ) ).

tff(fact_5054_summable__Leibniz_H_I4_J,axiom,
    ! [N_1: nat,A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),one_one_nat))))) ) ) ) ).

tff(fact_5055_cnj_OLIMSEQ,axiom,
    ! [X_3: fun_nat_complex,A: complex] :
      ( hBOOL(tendsto_nat_complex(X_3,A,sequentially))
     => hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3),hAPP_complex_complex(cnj,A),sequentially)) ) ).

tff(fact_5056_Re_OLIMSEQ,axiom,
    ! [X_3: fun_nat_complex,A: complex] :
      ( hBOOL(tendsto_nat_complex(X_3,A,sequentially))
     => hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3),hAPP_complex_real(re,A),sequentially)) ) ).

tff(fact_5057_Im_OLIMSEQ,axiom,
    ! [X_3: fun_nat_complex,A: complex] :
      ( hBOOL(tendsto_nat_complex(X_3,A,sequentially))
     => hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3),hAPP_complex_real(im,A),sequentially)) ) ).

tff(fact_5058_LIMSEQ__Complex,axiom,
    ! [Y_3: fun_nat_real,B: real,X_3: fun_nat_real,A: real] :
      ( hBOOL(tendsto_nat_real(X_3,A,sequentially))
     => ( hBOOL(tendsto_nat_real(Y_3,B,sequentially))
       => hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex_1),X_3)),Y_3),hAPP_real_complex(hAPP_r265291036omplex(complex_1,A),B),sequentially)) ) ) ).

tff(fact_5059_Bolzano__nest__unique,axiom,
    ! [P_1: fun_Pr1127657839l_bool,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))
         => ( hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B))),zero_zero_real,sequentially))
           => ? [L_1: real] :
                ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),L_1))
                & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially))
                & ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L_1),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))
                & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially)) ) ) ) ) ) ).

tff(fact_5060_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
         => hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)) ) ) ) ).

tff(fact_5061_summable__Leibniz_H_I2_J,axiom,
    ! [N_1: nat,A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)))) ) ) ) ).

tff(fact_5062_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(A))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N_2: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))))) ) ) ) ).

tff(fact_5063_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( hBOOL(monoseq_real(A))
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(A,zero_zero_nat)))
         => ! [N_2: nat] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))))) ) ) ) ).

tff(fact_5064_Sup__atLeastAtMost,axiom,
    ! [Y_2: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1))
     => ( comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Y_2),X_1)) = X_1 ) ) ).

tff(fact_5065_summable,axiom,
    ! [A: fun_nat_real] :
      ( hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))
       => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))
         => hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))) ) ) ) ).

tff(fact_5066_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U))) ).

tff(fact_5067_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) ).

tff(fact_5068_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
     => hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_1),Y_2))) ) ).

tff(fact_5069_all__nat__less,axiom,
    ! [P_1: fun_nat_bool,N_1: nat] :
      ( ! [M_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
         => hBOOL(hAPP_nat_bool(P_1,M_1)) )
    <=> ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N_1)))
         => hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_5070_ex__nat__less,axiom,
    ! [P_1: fun_nat_bool,N_1: nat] :
      ( ? [M_1: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
          & hBOOL(hAPP_nat_bool(P_1,M_1)) )
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N_1)))
          & hBOOL(hAPP_nat_bool(P_1,X_2)) ) ) ).

tff(fact_5071_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
    ! [Ma: nat,N_1: nat] : hAPP_n1699378549t_bool(ord_at238088361st_nat(Ma),N_1) = image_int_nat(nat_1,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1))) ).

tff(fact_5072_image__Suc__atLeastAtMost,axiom,
    ! [I_1: nat,J: nat] : image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) ).

tff(fact_5073_image__add__atLeastAtMost,axiom,
    ! [K_1: nat,I_1: nat,J: nat] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).

tff(fact_5074_fact__altdef__nat,axiom,
    ! [N_1: nat] : hAPP_nat_nat(fact_fact_nat,N_1) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),N_1)) ).

tff(fact_5075_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hAPP_n1699378549t_bool(ord_at4362885an_nat(L),hAPP_nat_nat(suc,U)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U) ).

tff(fact_5076_Re_Ocont,axiom,
    ! [A: complex] : hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,A),at_complex(A))) ).

tff(fact_5077_Im_Ocont,axiom,
    ! [A: complex] : hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,A),at_complex(A))) ).

tff(fact_5078_cnj_Ocont,axiom,
    ! [A: complex] : hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,A),at_complex(A))) ).

tff(fact_5079_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,U)),L) ).

tff(fact_5080_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hAPP_i1948725293t_bool(ord_at641636577an_int(L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),one_one_int)) = hAPP_i1948725293t_bool(ord_at875362053st_int(L),U) ).

tff(fact_5081_setsum__shift__lb__Suc0__0,axiom,
    ! [K_1: nat,F: fun_nat_nat] :
      ( ( hAPP_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,zero_zero_nat)),K_1)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K_1)) ) ) ).

tff(fact_5082_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)) = hAPP_int_nat(nat_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

tff(fact_5083_fact__div__fact,axiom,
    ! [N_1: nat,Ma: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(fact_fact_nat,Ma)),hAPP_nat_nat(fact_fact_nat,N_1)) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),Ma)) ) ) ).

tff(fact_5084_fact__altdef__int,axiom,
    ! [N_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1))
     => ( hAPP_int_int(fact_fact_int,N_1) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),N_1)) ) ) ).

tff(fact_5085_LIM__cos__div__sin,axiom,
    hBOOL(tendsto_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),cos)),sin),zero_zero_real,at_real(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).

tff(fact_5086_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( hBOOL(tendsto_real_real(F,L,at_real(C)))
     => ( ( L != zero_zero_real )
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ! [X_2: real] :
                ( ( ( X_2 != C )
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) )
               => ( hAPP_real_real(F,X_2) != zero_zero_real ) ) ) ) ) ).

tff(fact_5087_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( hBOOL(tendsto_real_real(F,L,at_real(C)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L))
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ! [X_2: real] :
                ( ( ( X_2 != C )
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) )
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X_2))) ) ) ) ) ).

tff(fact_5088_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( hBOOL(tendsto_real_real(F,L,at_real(C)))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real))
       => ? [R_1: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))
            & ! [X_2: real] :
                ( ( ( X_2 != C )
                  & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) )
               => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),zero_zero_real)) ) ) ) ) ).

tff(fact_5089_aset_I6_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Ta))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta)) ) ) ) ) ).

tff(fact_5090_aset_I8_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) ) ) ) ).

tff(fact_5091_bset_I6_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Ta))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta)) ) ) ) ).

tff(fact_5092_bset_I8_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X_2))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) ) ) ) ) ).

tff(fact_5093_aset_I7_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) ) ) ) ).

tff(fact_5094_bset_I5_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Ta))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta)) ) ) ) ).

tff(fact_5095_bset_I3_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( ( X_2 = Ta )
             => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1) = Ta ) ) ) ) ) ).

tff(fact_5096_aset_I3_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( ( X_2 = Ta )
             => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1) = Ta ) ) ) ) ) ).

tff(fact_5097_aset_I4_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( ( X_2 != Ta )
             => ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1) != Ta ) ) ) ) ) ).

tff(fact_5098_bset_I4_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( ( X_2 != Ta )
             => ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1) != Ta ) ) ) ) ) ).

tff(fact_5099_aset_I5_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Ta))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta)) ) ) ) ) ).

tff(fact_5100_bset_I7_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1))
       => ! [X_2: int] :
            ( ! [Xa: int] :
                ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
               => ! [Xb: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                   => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_2))
             => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) ) ) ) ) ).

tff(fact_5101_periodic__finite__ex,axiom,
    ! [P_1: fun_int_bool,D: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X_2: int,K: int] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
          <=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) )
       => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
        <=> ? [X_2: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
              & hBOOL(hAPP_int_bool(P_1,X_2)) ) ) ) ) ).

tff(fact_5102_aset_I10_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
           => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta))) ) ) ) ).

tff(fact_5103_bset_I10_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
           => ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta))) ) ) ) ).

tff(fact_5104_bset_I9_J,axiom,
    ! [Ta: int,B_1: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta))) ) ) ) ).

tff(fact_5105_aset_I9_J,axiom,
    ! [Ta: int,A_1: fun_int_bool,D: int,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1))
     => ! [X_2: int] :
          ( ! [Xa: int] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
             => ! [Xb: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                 => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta)))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta))) ) ) ) ).

tff(fact_5106_LIMSEQ__inverse__zero,axiom,
    ! [X_3: fun_nat_real] :
      ( ! [R_1: real] :
        ? [N_3: nat] :
        ! [N_2: nat] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_3,N_2))) )
     => hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_3),zero_zero_real,sequentially)) ) ).

tff(fact_5107_real__scaleR__def,axiom,
    ! [A_3: real,X: real] : hAPP_real_real(scaleR_scaleR_real(A_3),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),X) ).

tff(fact_5108_complex__Im__scaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X)) ).

tff(fact_5109_complex__Re__scaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X)) ).

tff(fact_5110_complex__scaleR,axiom,
    ! [R_3: real,A_3: real,B_2: real] : hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_real_complex(hAPP_r265291036omplex(complex_1,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),B_2)) ).

tff(fact_5111_Re_OscaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(re,X)) ).

tff(fact_5112_Im_OscaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(im,X)) ).

tff(fact_5113_cnj_OscaleR,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_complex_complex(cnj,X)) ).

tff(fact_5114_complex__sgn__def,axiom,
    ! [X: complex] : hAPP_complex_complex(sgn_sgn_complex,X) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X))),X) ).

tff(fact_5115_complex__scaleR__def,axiom,
    ! [R_3: real,X: complex] : hAPP_complex_complex(scaleR1652505878omplex(R_3),X) = hAPP_real_complex(hAPP_r265291036omplex(complex_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X))) ).

tff(fact_5116_trivial__limit__sequentially,axiom,
    ~ hBOOL(trivial_limit_nat(sequentially)) ).

tff(fact_5117_pred__nat__def,axiom,
    pred_nat = collec1979865426at_nat(produc1922581855t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat)),suc))) ).

tff(fact_5118_gcd__coprime__exists__int,axiom,
    ! [A_3: int,B_2: int] :
      ( ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int )
     => ? [A_4: int,B_3: int] :
          ( ( A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_4),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) )
          & ( B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) )
          & ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),B_3) = one_one_int ) ) ) ).

tff(fact_5119_min__number__of__Suc,axiom,
    ! [N_1: nat,V: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_1),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) ).

tff(fact_5120_min__Suc__number__of,axiom,
    ! [V: int,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N_1))) ).

tff(fact_5121_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

tff(fact_5122_min__0R,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

tff(fact_5123_min__0L,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

tff(fact_5124_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)) ).

tff(fact_5125_min__Suc2,axiom,
    ! [Ma: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Ma),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_min_nat),N_1))),Ma) ).

tff(fact_5126_min__Suc1,axiom,
    ! [N_1: nat,Ma: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N_1)),Ma) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_min_nat,N_1))),Ma) ).

tff(fact_5127_vanishes__mult__bounded,axiom,
    ! [Y_3: fun_nat_rat,X_3: fun_nat_rat] :
      ( ? [A_2: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_2))
          & ! [N_2: nat] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),A_2)) )
     => ( hBOOL(vanishes(Y_3))
       => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,times_times_rat),X_3)),Y_3))) ) ) ).

tff(fact_5128_SetInterval_Otransfer__int__nat__set__functions,axiom,
    ! [N_1: int,Ma: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,Ma))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,N_1))
       => ( hAPP_i1948725293t_bool(ord_at875362053st_int(Ma),N_1) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat_1,Ma)),hAPP_int_nat(nat_1,N_1))) ) ) ) ).

tff(fact_5129_is__nat__def,axiom,
    ! [X_1: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).

tff(fact_5130_vanishes__diff,axiom,
    ! [Y_3: fun_nat_rat,X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => ( hBOOL(vanishes(Y_3))
       => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,minus_minus_rat),X_3)),Y_3))) ) ) ).

tff(fact_5131_vanishes__add,axiom,
    ! [Y_3: fun_nat_rat,X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => ( hBOOL(vanishes(Y_3))
       => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,plus_plus_rat),X_3)),Y_3))) ) ) ).

tff(fact_5132_vanishes__const,axiom,
    ! [C: rat] :
      ( hBOOL(vanishes(cOMBK_rat_nat(C)))
    <=> ( C = zero_zero_rat ) ) ).

tff(fact_5133_vanishes__minus,axiom,
    ! [X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_3))) ) ).

tff(fact_5134_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

tff(fact_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
    ! [N: nat,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).

tff(fact_5136_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X),Y))) ) ) ).

tff(fact_5137_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))) ) ) ).

tff(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(nat_tsub(X),Y))) ) ) ).

tff(fact_5139_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
    ! [X_1: int,A_1: fun_int_bool] :
      ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1))
       => hBOOL(hAPP_int_bool(nat_is_nat,X_1)) ) ) ).

tff(fact_5140_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,zero_zero_int)) ).

tff(fact_5141_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,one_one_int)) ).

tff(fact_5142_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

tff(fact_5143_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
    ! [Z_1: nat] : hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(semiri1621563631at_int,Z_1))) ).

tff(fact_5144_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
    ! [Y_2: int,X_1: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
     => hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_1),Y_2))) ) ).

tff(fact_5145_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) ).

tff(fact_5146_transfer__int__nat__gcd__closures_I1_J,axiom,
    ! [Y: int,X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ).

tff(fact_5147_transfer__int__nat__factorial__closure,axiom,
    ! [X: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))) ) ).

tff(fact_5148_transfer__int__nat__set__relations_I2_J,axiom,
    ! [A_1: fun_int_bool,X_1: int] :
      ( hBOOL(hAPP_int_bool(nat_is_nat,X_1))
     => ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1))
        <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat_1,X_1)),image_int_nat(nat_1,A_1))) ) ) ) ).

tff(fact_5149_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
    hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).

tff(fact_5150_vanishes__def,axiom,
    ! [X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
    <=> ! [R_1: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
         => ? [K: nat] :
            ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)) ) ) ) ).

tff(fact_5151_vanishesD,axiom,
    ! [R_2: rat,X_3: fun_nat_rat] :
      ( hBOOL(vanishes(X_3))
     => ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2))
       => ? [K: nat] :
          ! [N_2: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
           => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_2)) ) ) ) ).

tff(fact_5152_vanishesI,axiom,
    ! [X_3: fun_nat_rat] :
      ( ! [R_1: rat] :
          ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
         => ? [K: nat] :
            ! [N_2: nat] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2))
             => hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)) ) )
     => hBOOL(vanishes(X_3)) ) ).

tff(fact_5153_zcongm__def,axiom,
    ! [Ma: int,X_2: int,Xa: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X_2),Xa))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),Xa),Ma)) ) ).

tff(fact_5154_cpmi,axiom,
    ! [B_1: fun_int_bool,P_2: fun_int_bool,P_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ? [Z: int] :
          ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z))
           => ( hBOOL(hAPP_int_bool(P_1,X_2))
            <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
       => ( ! [X_2: int] :
              ( ! [Xa: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                 => ! [Xb: int] :
                      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1))
                     => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) ) )
         => ( ! [X_2: int,K: int] :
                ( hBOOL(hAPP_int_bool(P_2,X_2))
              <=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) )
           => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
            <=> ( ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & hBOOL(hAPP_int_bool(P_2,X_2)) )
                | ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & ? [Xa: int] :
                        ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),B_1))
                        & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),X_2))) ) ) ) ) ) ) ) ) ).

tff(fact_5155_bijzcong__zcong__prod,axiom,
    ! [A_1: fun_int_bool,B_1: fun_int_bool,Ma: int] :
      ( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_1)),bijR_int_int(zcongm(Ma))))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A_1)),big_co1548731110nt_int(cOMBI_int,B_1)),Ma)) ) ).

tff(fact_5156_cppi,axiom,
    ! [A_1: fun_int_bool,P_2: fun_int_bool,P_1: fun_int_bool,D_1: int] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ? [Z: int] :
          ! [X_2: int] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2))
           => ( hBOOL(hAPP_int_bool(P_1,X_2))
            <=> hBOOL(hAPP_int_bool(P_2,X_2)) ) )
       => ( ! [X_2: int] :
              ( ! [Xa: int] :
                  ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                 => ! [Xb: int] :
                      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1))
                     => ( X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
             => ( hBOOL(hAPP_int_bool(P_1,X_2))
               => hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) ) )
         => ( ! [X_2: int,K: int] :
                ( hBOOL(hAPP_int_bool(P_2,X_2))
              <=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) )
           => ( ? [X1: int] : hBOOL(hAPP_int_bool(P_1,X1))
            <=> ( ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & hBOOL(hAPP_int_bool(P_2,X_2)) )
                | ? [X_2: int] :
                    ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
                    & ? [Xa: int] :
                        ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1))
                        & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),X_2))) ) ) ) ) ) ) ) ) ).

tff(fact_5157_increasing__LIMSEQ,axiom,
    ! [L: real,F: fun_nat_real] :
      ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))
     => ( ! [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),L))
       => ( ! [E: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
             => ? [N_2: nat] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,N_2)),E))) )
         => hBOOL(tendsto_nat_real(F,L,sequentially)) ) ) ) ).

tff(fact_5158_isCont__ln,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
     => hBOOL(hAPP_real_bool(isCont_real_real(ln),X_1)) ) ).

tff(fact_5159_isCont__real__root,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),X_1)) ) ).

tff(fact_5160_isCont__tan,axiom,
    ! [X_1: real] :
      ( ( hAPP_real_real(cos,X_1) != zero_zero_real )
     => hBOOL(hAPP_real_bool(isCont_real_real(tan),X_1)) ) ).

tff(fact_5161_isCont__cos,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(cos),X_1)) ).

tff(fact_5162_isCont__arctan,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(arctan),X_1)) ).

tff(fact_5163_isCont__real__sqrt,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(sqrt),X_1)) ).

tff(fact_5164_isCont__sin,axiom,
    ! [X_1: real] : hBOOL(hAPP_real_bool(isCont_real_real(sin),X_1)) ).

tff(fact_5165_cnj_OisCont,axiom,
    ! [A: complex] : hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),A)) ).

tff(fact_5166_Im_OisCont,axiom,
    ! [A: complex] : hBOOL(hAPP_complex_bool(isCont_complex_real(im),A)) ).

tff(fact_5167_Re_OisCont,axiom,
    ! [A: complex] : hBOOL(hAPP_complex_bool(isCont_complex_real(re),A)) ).

tff(fact_5168_isCont__abs,axiom,
    ! [A: real] : hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),A)) ).

tff(fact_5169_isCont__root__zero,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),zero_zero_real)) ) ).

tff(fact_5170_isCont__root__neg,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real))
       => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),X_1)) ) ) ).

tff(fact_5171_isCont__root__pos,axiom,
    ! [X_1: real,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))
       => hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),X_1)) ) ) ).

tff(fact_5172_isCont__arccos,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(isCont_real_real(arccos),X_1)) ) ) ).

tff(fact_5173_isCont__arcsin,axiom,
    ! [X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1))
     => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))
       => hBOOL(hAPP_real_bool(isCont_real_real(arcsin),X_1)) ) ) ).

tff(fact_5174_LIM__less__bound,axiom,
    ! [F: fun_real_real,B: real,X_1: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),X_1))
     => ( ! [X_2: real] :
            ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(B),X_1)))
           => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_2))) )
       => ( hBOOL(hAPP_real_bool(isCont_real_real(F),X_1))
         => hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_1))) ) ) ) ).

tff(fact_5175_DERIV__inverse__function,axiom,
    ! [B: real,A: real,F: fun_real_real,G: fun_real_real,X_1: real,D_1: real] :
      ( hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,X_1)),D_1))
     => ( ( D_1 != zero_zero_real )
       => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_1))
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B))
           => ( ! [Y_1: real] :
                  ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Y_1))
                    & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B)) )
                 => ( hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1 ) )
             => ( hBOOL(hAPP_real_bool(isCont_real_real(G),X_1))
               => hBOOL(hAPP_real_bool(deriv_real(G,X_1),hAPP_real_real(inverse_inverse_real,D_1))) ) ) ) ) ) ) ).

tff(fact_5176_isCont__inv__fun,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,X_1))) ) ) ) ).

tff(fact_5177_DERIV__isconst2,axiom,
    ! [X_1: real,F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) )
       => ( ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) )
             => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
         => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_1))
           => ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B))
             => ( hAPP_real_real(F,X_1) = hAPP_real_real(F,A) ) ) ) ) ) ) ).

tff(fact_5178_DERIV__isconst__end,axiom,
    ! [F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) )
       => ( ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) )
             => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
         => ( hAPP_real_real(F,B) = hAPP_real_real(F,A) ) ) ) ) ).

tff(fact_5179_DERIV__isconst1,axiom,
    ! [F: fun_real_real,A: real,B: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))
     => ( ! [X_2: real] :
            ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
           => hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)) )
       => ( ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) )
             => hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)) )
         => ! [X_2: real] :
              ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2))
                & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) )
             => ( hAPP_real_real(F,X_2) = hAPP_real_real(F,A) ) ) ) ) ) ).

tff(fact_5180_isCont__inv__fun__inv,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => ? [E: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
              & ! [Y_1: real] :
                  ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,X_1)))))
                    & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,X_1)))),E)) )
                 => ( hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1 ) ) ) ) ) ) ).

tff(fact_5181_lemma__isCont__inj2,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => ? [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,X_1))) ) ) ) ) ).

tff(fact_5182_lemma__isCont__inj,axiom,
    ! [G: fun_real_real,F: fun_real_real,X_1: real,D: real] :
      ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
           => ( hAPP_real_real(G,hAPP_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
             => hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
         => ? [Z: real] :
              ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D))
              & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,Z))) ) ) ) ) ).

tff(fact_5183_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [N_1: nat] : hAPP_n1699378549t_bool(ord_atMost_nat,N_1) = image_int_nat(nat_1,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) ).

tff(fact_5184_max__number__of__Suc,axiom,
    ! [N_1: nat,V: int] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),hAPP_int_nat(nat_1,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) ).

tff(fact_5185_finite__atMost,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K_1))) ).

tff(fact_5186_atLeast0AtMost,axiom,
    ! [N_1: nat] : hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N_1) = hAPP_n1699378549t_bool(ord_atMost_nat,N_1) ).

tff(fact_5187_lessThan__Suc__atMost,axiom,
    ! [K_1: nat] : hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K_1)) = hAPP_n1699378549t_bool(ord_atMost_nat,K_1) ).

tff(fact_5188_max__Suc1,axiom,
    ! [N_1: nat,Ma: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N_1)),Ma) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N_1),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,N_1))),Ma) ).

tff(fact_5189_max__Suc2,axiom,
    ! [Ma: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N_1),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_max_nat),N_1))),Ma) ).

tff(fact_5190_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),M) ).

tff(fact_5191_max__0R,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

tff(fact_5192_max__0L,axiom,
    ! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

tff(fact_5193_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N)) ).

tff(fact_5194_card__atMost,axiom,
    ! [U: nat] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U)) = hAPP_nat_nat(suc,U) ).

tff(fact_5195_finite__nat__iff__bounded__le,axiom,
    ! [S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
    <=> ? [K: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K))) ) ).

%----Helper facts (224)
tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fNot,P)) ) ).

tff(help_fTrue_1_1_U,axiom,
    hBOOL(fTrue) ).

tff(help_fTrue_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(P) ) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(Q) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | hBOOL(fdisj(P,Q)) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(fdisj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(fdisj(P,Q))
      | hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ hBOOL(fFalse) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fimplies(P),Q)) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(fimplies(P),Q)) ) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fimplies(P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_If_1_1_If_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__Int__Oint_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fEx_1_1_fEx_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_bool,X: int] :
      ( ~ hBOOL(hAPP_int_bool(P,X))
      | hBOOL(hAPP_f448129468l_bool(fEx_int,P)) ) ).

tff(help_fEx_1_1_fEx_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ hBOOL(hAPP_nat_bool(P,X))
      | hBOOL(hAPP_f54304608l_bool(fEx_nat,P)) ) ).

tff(help_COMBI_1_1_COMBI_000tc__Int__Oint_U,axiom,
    ! [P: int] : hAPP_int_int(cOMBI_int,P) = P ).

tff(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
    ! [P: nat] : hAPP_nat_nat(cOMBI_nat,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)) ) ).

tff(help_If_1_1_If_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__RealDef__Oreal_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBI_1_1_COMBI_000tc__RealDef__Oreal_U,axiom,
    ! [P: real] : hAPP_real_real(cOMBI_real,P) = P ).

tff(help_fequal_1_1_fequal_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__Rat__Orat_000tc__Nat__Onat_U,axiom,
    ! [P: rat,Q: nat] : hAPP_nat_rat(cOMBK_rat_nat(P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: bool,Q: nat] : hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ).

tff(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_int,Q: fun_int_int,R: int] : hAPP_int_int(hAPP_f1356864277nt_int(hAPP_f747162069nt_int(cOMBB_int_int_int,P),Q),R) = hAPP_int_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_int,Q: fun_nat_int,R: nat] : hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R) = hAPP_int_int(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Rat__Orat_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_rat,Q: fun_int_int,R: int] : hAPP_int_rat(hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,P),Q),R) = hAPP_int_rat(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__Rat__Orat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_rat_rat,Q: fun_nat_rat,R: nat] : hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,P),Q),R) = hAPP_rat_rat(P,hAPP_nat_rat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_fun_int_int,Q: int,R: int] : hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R) = hAPP_int_int(hAPP_int_fun_int_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_fun_nat_int,Q: nat,R: int] : hAPP_int_int(hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,P),Q),R) = hAPP_nat_int(hAPP_int_fun_nat_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U,axiom,
    ! [P: fun_nat_fun_int_int,Q: int,R: nat] : hAPP_nat_int(hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,P),Q),R) = hAPP_int_int(hAPP_nat_fun_int_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: nat,R: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat_U,axiom,
    ! [P: fun_nat_fun_rat_rat,Q: fun_nat_rat,R: nat] : hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,P),Q),R) = hAPP_rat_rat(hAPP_nat_fun_rat_rat(P,R),hAPP_nat_rat(Q,R)) ).

tff(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
    ! [X: produc914805421l_real,Y: produc914805421l_real] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fTrue),X),Y) = X ).

tff(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
    ! [X: produc914805421l_real,Y: produc914805421l_real] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X),Y) = Y ).

tff(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_bool,Q: fun_int_int,R: int] : hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R) = hAPP_int_bool(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_int_bool,Q: fun_nat_int,R: nat] : hAPP_nat_bool(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P),Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_nat_bool,Q: fun_int_nat,R: int] : hAPP_int_bool(hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P),Q),R) = hAPP_nat_bool(P,hAPP_int_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_rat_bool,Q: fun_int_rat,R: int] : hAPP_int_bool(hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,P),Q),R) = hAPP_rat_bool(P,hAPP_int_rat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_int_bool,Q: int,R: int] : hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_int_fun_rat_bool,Q: rat,R: int] : hAPP_int_bool(hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,P),Q),R) = hAPP_rat_bool(hAPP_i16292313t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_bool_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_in531499254l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__Int__Oint_U,axiom,
    ! [P: fun_int_real,Q: fun_int_int,R: int] : hAPP_int_real(hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,P),Q),R) = hAPP_int_real(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Nat__Onat_U,axiom,
    ! [P: fun_nat_real,Q: fun_nat_nat,R: nat] : hAPP_nat_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,P),Q),R) = hAPP_nat_real(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P: fun_real_bool,Q: fun_int_real,R: int] : hAPP_int_bool(hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,P),Q),R) = hAPP_real_bool(P,hAPP_int_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_real_bool,Q: fun_nat_real,R: nat] : hAPP_nat_bool(hAPP_f737665877t_bool(hAPP_f605995867t_bool(cOMBB_real_bool_nat,P),Q),R) = hAPP_real_bool(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__HOL__Obool_U,axiom,
    ! [P: fun_in1251377736l_bool,Q: real,R: int] : hAPP_int_bool(hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,P),Q),R) = hAPP_real_bool(hAPP_i688218208l_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_re1114994977l_bool,Q: fun_real_bool,R: real] : hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Int__Oin,axiom,
    ! [P: fun_real_real,Q: fun_int_real,R: int] : hAPP_int_real(hAPP_f269654879t_real(hAPP_f766218899t_real(cOMBB_real_real_int,P),Q),R) = hAPP_real_real(P,hAPP_int_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona,axiom,
    ! [P: fun_real_real,Q: fun_nat_real,R: nat] : hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R) = hAPP_real_real(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea,axiom,
    ! [P: fun_na182379510l_real,Q: real,R: nat] : hAPP_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__Nat__Onat_000tc__RealDef__Orea,axiom,
    ! [P: fun_re1927296998t_real,Q: nat,R: real] : hAPP_real_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,P),Q),R) = hAPP_nat_real(hAPP_r474017924t_real(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: nat,Q: fun_int_bool] : hAPP_f957591787ol_nat(cOMBK_1643584600t_bool(P),Q) = P ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea,axiom,
    ! [P: fun_na182379510l_real,Q: fun_nat_real,R: nat] : hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__RealDef__Ore,axiom,
    ! [P: fun_real_bool,Q: fun_real_real,R: real] : hAPP_real_bool(hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,P),Q),R) = hAPP_real_bool(P,hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL__Obo,axiom,
    ! [P: fun_re413263731l_bool,Q: real,R: real] : hAPP_real_bool(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,P),Q),R) = hAPP_real_bool(hAPP_r1134773055l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__,axiom,
    ! [P: fun_complex_real,Q: fun_nat_complex,R: nat] : hAPP_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,P),Q),R) = hAPP_complex_real(P,hAPP_nat_complex(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocom,axiom,
    ! [P: fun_na1566252557omplex,Q: fun_nat_real,R: nat] : hAPP_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,P),Q),R) = hAPP_real_complex(hAPP_n1983812447omplex(P,R),hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
    ! [P: fun_real_real,Q: fun_real_real,R: real] : hAPP_real_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,P),Q),R) = hAPP_real_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
    ! [P: fun_re229243589l_real,Q: real,R: real] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_,axiom,
    ! [P: fun_re229243589l_real,Q: fun_real_real,R: real] : hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Na,axiom,
    ! [P: fun_complex_complex,Q: fun_nat_complex,R: nat] : hAPP_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,P),Q),R) = hAPP_complex_complex(P,hAPP_nat_complex(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_int_int,R: int] : hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_458,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_nat_int,R: nat] : hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,axiom,
    ! [P: fun_nat_fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R) = hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__Rat__Orat_J,axiom,
    ! [P: fun_rat_fun_rat_rat,Q: fun_nat_rat,R: nat] : hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),Q),R) = hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin,axiom,
    ! [P: fun_int_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),Q),R) = hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
    ! [P: fun_in2003698982nt_int,Q: fun_int_int,R: int] : hAPP_i1524277240nt_int(hAPP_f883232462nt_int(hAPP_f2002277285nt_int(cOMBB_1846624359nt_int,P),Q),R) = hAPP_i1524277240nt_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_rat_fun_rat_bool,Q: fun_int_rat,R: int] : hAPP_i16292313t_bool(hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,P),Q),R) = hAPP_r2115560837t_bool(P,hAPP_int_rat(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
    ! [P: fun_in351380826t_bool,Q: int,R: int] : hAPP_i1948725293t_bool(hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,P),Q),R) = hAPP_i1948725293t_bool(hAPP_i345030060t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin,axiom,
    ! [P: fun_in1111845413nt_int,Q: int,R: int] : hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,P),Q),R) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin,axiom,
    ! [P: fun_Pr2134412615nt_int,Q: int,R: product_prod_int_int] : hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin,axiom,
    ! [P: fun_Pr2134412615nt_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_459,axiom,
    ! [P: fun_Pr1826394355nt_rat,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1853363688nt_rat(hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,P),Q),R) = hAPP_int_rat(hAPP_P507092031nt_rat(P,R),hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi,axiom,
    ! [P: fun_int_bool,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,P),Q),R) = hAPP_int_bool(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi,axiom,
    ! [P: fun_rat_bool,Q: fun_Pr1024438428nt_rat,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,P),Q),R) = hAPP_rat_bool(P,hAPP_P1853363688nt_rat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu819804446l_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu425979586l_bool,Q: fun_int_fun_nat_bool,R: int] : hAPP_int_bool(hAPP_f41603166t_bool(hAPP_f389116883t_bool(cOMBB_1593174431ol_int,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in938817141l_bool,Q: fun_int_bool,R: int] : hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_460,axiom,
    ! [P: fun_Pr1149636328t_bool,Q: int,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_461,axiom,
    ! [P: fun_Pr1149636328t_bool,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f1325804733t_bool(cOMBS_480885903t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__prod_Itc__Int__O,axiom,
    ! [P: fun_bool_bool,Q: fun_Pr974702441t_bool,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,P),Q),R) = hAPP_bool_bool(P,hAPP_P603027463t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_int_bool,R: int] : hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_462,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obo,axiom,
    ! [P: fun_Pr189909033l_bool,Q: fun_Pr974702441t_bool,R: product_prod_int_int] : hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,P),Q),R) = hAPP_bool_bool(hAPP_P921862901l_bool(P,R),hAPP_P603027463t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__RealDef_,axiom,
    ! [P: fun_fu95670548l_real,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,P),Q),R) = hAPP_f1406902706l_real(P,hAPP_n1699378549t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo_463,axiom,
    ! [P: fun_fu819804446l_bool,Q: fun_re546217968t_bool,R: real] : hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_464,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_real_bool,R: real] : hAPP_r1938957037l_bool(hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_real_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDe,axiom,
    ! [P: fun_na182379510l_real,Q: fun_nat_nat,R: nat] : hAPP_n546711566l_real(hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,P),Q),R) = hAPP_n546711566l_real(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H,axiom,
    ! [P: fun_re413263731l_bool,Q: fun_int_real,R: int] : hAPP_i688218208l_bool(hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_int_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O,axiom,
    ! [P: fun_re1879328022l_real,Q: fun_nat_bool,R: real] : hAPP_real_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,P),Q),R) = hAPP_f1406902706l_real(hAPP_r1409264866l_real(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__RealDe,axiom,
    ! [P: fun_re1927296998t_real,Q: fun_real_real,R: real] : hAPP_r474017924t_real(hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,P),Q),R) = hAPP_r474017924t_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
    ! [P: fun_re229243589l_real,Q: fun_int_real,R: int] : hAPP_i371647666l_real(hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_int_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_465,axiom,
    ! [P: fun_re229243589l_real,Q: fun_nat_real,R: nat] : hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__Real,axiom,
    ! [P: fun_fu1122561126l_real,Q: fun_re1927296998t_real,R: real] : hAPP_real_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,P),Q),R) = hAPP_f352196356l_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real,axiom,
    ! [P: fun_na768876520l_real,Q: real,R: nat] : hAPP_n546711566l_real(hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real,axiom,
    ! [P: fun_na768876520l_real,Q: fun_nat_real,R: nat] : hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H_466,axiom,
    ! [P: fun_re413263731l_bool,Q: fun_real_real,R: real] : hAPP_r1134773055l_bool(hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc,axiom,
    ! [P: fun_Pr608224961l_real,Q: fun_na1745536976l_real,R: nat] : hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R) = hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_467,axiom,
    ! [P: fun_Pr1127657839l_bool,Q: fun_re785671361l_real,R: real] : hAPP_real_bool(hAPP_f330985925l_bool(hAPP_f940061907l_bool(cOMBB_1795223523l_real,P),Q),R) = hAPP_P1333854989l_bool(P,hAPP_r1195171167l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__C,axiom,
    ! [P: fun_re398676606omplex,Q: fun_nat_real,R: nat] : hAPP_n1983812447omplex(hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,P),Q),R) = hAPP_r265291036omplex(P,hAPP_nat_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_468,axiom,
    ! [P: fun_re229243589l_real,Q: fun_real_real,R: real] : hAPP_r1250527377l_real(hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__fun_Itc_,axiom,
    ! [P: fun_re1227808089l_real,Q: real,R: real] : hAPP_r1250527377l_real(hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_r337325687l_real(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__,axiom,
    ! [P: fun_re785671361l_real,Q: fun_real_real,R: real] : hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f800361423l_real(cOMBB_2107848501l_real,P),Q),R) = hAPP_r1195171167l_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc,axiom,
    ! [P: fun_re1550493891l_real,Q: fun_real_real,R: real] : hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f1730463541l_real(cOMBS_511527878l_real,P),Q),R) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(P,R),hAPP_real_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,axiom,
    ! [P: fun_in1797533524nt_int,Q: fun_int_int,R: int] : hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,P),Q),R) = hAPP_f1356864277nt_int(hAPP_i744251628nt_int(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_469,axiom,
    ! [P: fun_int_fun_int_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P2110489235nt_int(hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J,axiom,
    ! [P: fun_int_fun_int_rat,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P507092031nt_rat(hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,P),Q),R) = hAPP_int_fun_int_rat(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I,axiom,
    ! [P: fun_in1111845413nt_int,Q: fun_int_int,R: int] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,P),Q),R) = hAPP_i1584592887nt_int(P,hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__prod_Itc__N,axiom,
    ! [P: fun_na257798253at_nat,Q: fun_nat_nat,R: nat] : hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),Q),R) = hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__,axiom,
    ! [P: fun_fu205926450t_bool,Q: fun_int_fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,P),Q),R) = hAPP_f429935748t_bool(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_470,axiom,
    ! [P: fun_in67169585t_bool,Q: fun_int_int,R: int] : hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,P),Q),R) = hAPP_f429935748t_bool(hAPP_i872162435t_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint,axiom,
    ! [P: fun_in865885604nt_int,Q: fun_int_int,R: int] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,P),Q),R) = hAPP_i1584592887nt_int(hAPP_i1778081398nt_int(P,R),hAPP_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__471,axiom,
    ! [P: fun_int_fun_int_bool,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P252494982t_bool(hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_nat,R: nat] : hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu554825757t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc,axiom,
    ! [P: fun_Pr241715507nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f151095827nt_int(cOMBB_1389251822nt_int,P),Q),R) = hAPP_P1975530577nt_int(P,hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__472,axiom,
    ! [P: fun_in719570932t_bool,Q: fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in319042876t_bool,Q: fun_nat_bool,R: int] : hAPP_i418383825t_bool(hAPP_f289318786t_bool(hAPP_f729588221t_bool(cOMBC_678732631t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_i1796122964t_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P: fun_in719570932t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
    ! [P: fun_in989261322nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f713949180nt_int(cOMBS_1891347093nt_int,P),Q),R) = hAPP_P1975530577nt_int(hAPP_i1216375074nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_473,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Pr974702441t_bool,R: product_prod_int_int] : hAPP_P921862901l_bool(hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_P603027463t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_000tc__fun_,axiom,
    ! [P: fun_fu2100483183t_bool,Q: fun_int_fun_int_real,R: int] : hAPP_i1948725293t_bool(hAPP_f351634443t_bool(hAPP_f1576199059t_bool(cOMBB_95956018ol_int,P),Q),R) = hAPP_f429384717t_bool(P,hAPP_i627165055t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun_,axiom,
    ! [P: fun_fu1532576183t_bool,Q: fun_int_fun_nat_real,R: int] : hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),Q),R) = hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Or,axiom,
    ! [P: fun_in1390825880t_real,Q: fun_int_real,R: int] : hAPP_i627165055t_real(hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,P),Q),R) = hAPP_f269654879t_real(hAPP_i1671359920t_real(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Or,axiom,
    ! [P: fun_in464278240t_real,Q: fun_nat_real,R: int] : hAPP_i157317923t_real(hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,P),Q),R) = hAPP_f1888375463t_real(hAPP_i447943416t_real(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__O,axiom,
    ! [P: fun_re573205052t_real,Q: fun_nat_nat,R: real] : hAPP_r474017924t_real(hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,P),Q),R) = hAPP_f1676298106t_real(hAPP_r50993370t_real(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__474,axiom,
    ! [P: fun_fu1569755401t_real,Q: fun_re1927296998t_real,R: real] : hAPP_r474017924t_real(hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,P),Q),R) = hAPP_f1888375463t_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_I,axiom,
    ! [P: fun_bo1998411783l_real,Q: fun_nat_bool,R: nat] : hAPP_n682674106l_real(hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,P),Q),R) = hAPP_b646336293l_real(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p,axiom,
    ! [P: fun_re1550493891l_real,Q: fun_real_real,R: real] : hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f2114902373l_real(cOMBB_237479429l_real,P),Q),R) = hAPP_r2019696015l_real(P,hAPP_real_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___475,axiom,
    ! [P: fun_fu215647537t_bool,Q: fun_int_fun_int_int,R: int] : hAPP_i345030060t_bool(hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,P),Q),R) = hAPP_f1791153283t_bool(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_476,axiom,
    ! [P: fun_in2078400507nt_int,Q: fun_int_int,R: int] : hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,P),Q),R) = hAPP_f883232462nt_int(hAPP_i998473677nt_int(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__,axiom,
    ! [P: fun_re923322561l_real,Q: fun_re785671361l_real,R: real] : hAPP_r1195171167l_real(hAPP_f1736103445l_real(hAPP_f1908301369l_real(cOMBS_902912708l_real,P),Q),R) = hAPP_P1860904029l_real(hAPP_r938550413l_real(P,R),hAPP_r1195171167l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__I,axiom,
    ! [P: fun_in1211676399nt_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,P),Q),R) = hAPP_i61245889nt_int(P,hAPP_P1175774780nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__477,axiom,
    ! [P: fun_fu610077323l_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R) = hAPP_f2144054103l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu2104074051l_bool,Q: fun_int_fun_nat_bool,R: int] : hAPP_i614188481l_bool(hAPP_f252743847l_bool(hAPP_f643658017l_bool(cOMBB_2046938128ol_int,P),Q),R) = hAPP_f1146629647l_bool(P,hAPP_i418383825t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL_,axiom,
    ! [P: fun_fu1221449756t_bool,Q: fun_re386138607t_bool,R: real] : hAPP_r481142414t_bool(hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,P),Q),R) = hAPP_f2119767738t_bool(P,hAPP_r632163725t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL_,axiom,
    ! [P: fun_fu1290768832t_bool,Q: fun_re2063775123t_bool,R: real] : hAPP_r481142414t_bool(hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,P),Q),R) = hAPP_f41603166t_bool(P,hAPP_r373117745t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__HOL__O,axiom,
    ! [P: fun_re1058169246t_bool,Q: fun_int_bool,R: real] : hAPP_r632163725t_bool(hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,P),Q),R) = hAPP_f1048215610t_bool(hAPP_r1623423338t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O_478,axiom,
    ! [P: fun_re507499238t_bool,Q: fun_nat_bool,R: real] : hAPP_r373117745t_bool(hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,P),Q),R) = hAPP_f289318786t_bool(hAPP_r653725362t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__479,axiom,
    ! [P: fun_fu705349683l_real,Q: fun_re1927296998t_real,R: real] : hAPP_r1409264866l_real(hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,P),Q),R) = hAPP_f1509287679l_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___480,axiom,
    ! [P: fun_fu1480724285nt_int,Q: fun_int_fun_int_int,R: int] : hAPP_i744251628nt_int(hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,P),Q),R) = hAPP_f747162069nt_int(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___481,axiom,
    ! [P: fun_fu1843361787nt_int,Q: fun_int_fun_int_int,R: int] : hAPP_i1778081398nt_int(hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,P),Q),R) = hAPP_f1574055885nt_int(P,hAPP_int_fun_int_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea,axiom,
    ! [P: fun_fu1280874925l_bool,Q: fun_re1550493891l_real,R: real] : hAPP_r1134773055l_bool(hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,P),Q),R) = hAPP_f330985925l_bool(P,hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__482,axiom,
    ! [P: fun_fu2112913523t_bool,Q: fun_int_fun_int_bool,R: int] : hAPP_i872162435t_bool(hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,P),Q),R) = hAPP_f1512942609t_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__483,axiom,
    ! [P: fun_fu257678363t_bool,Q: fun_in351380826t_bool,R: int] : hAPP_i345030060t_bool(hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,P),Q),R) = hAPP_f428220345t_bool(P,hAPP_i345030060t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I,axiom,
    ! [P: fun_in792564872nt_int,Q: fun_in1111845413nt_int,R: int] : hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R) = hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,axiom,
    ! [P: fun_in941977532nt_int,Q: product_prod_int_int,R: int] : hAPP_i1216375074nt_int(hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint__484,axiom,
    ! [P: fun_in941977532nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1216375074nt_int(hAPP_f1375227553nt_int(hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc,axiom,
    ! [P: fun_fu1909738875t_bool,Q: fun_in531499254l_bool,R: int] : hAPP_i1529485324t_bool(hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,P),Q),R) = hAPP_f627970963t_bool(P,hAPP_i68813070l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_,axiom,
    ! [P: fun_fu2120820641l_real,Q: fun_re229243589l_real,R: real] : hAPP_r337325687l_real(hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,P),Q),R) = hAPP_f556082547l_real(P,hAPP_r1250527377l_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
    ! [P: fun_re955532029l_real,Q: fun_real_real,R: real] : hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R,axiom,
    ! [P: fun_re955532029l_real,Q: fun_re229243589l_real,R: real] : hAPP_r2019696015l_real(hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),hAPP_r1250527377l_real(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It,axiom,
    ! [P: fun_re521366895t_bool,Q: fun_int_fun_int_real,R: real] : hAPP_r632163725t_bool(hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,P),Q),R) = hAPP_f351634443t_bool(hAPP_r1392521531t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It_485,axiom,
    ! [P: fun_re811280567t_bool,Q: fun_int_fun_nat_real,R: real] : hAPP_r373117745t_bool(hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,P),Q),R) = hAPP_f1222178131t_bool(hAPP_r2052264067t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__486,axiom,
    ! [P: fun_fu1719442527t_real,Q: fun_re1927296998t_real,R: real] : hAPP_r50993370t_real(hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,P),Q),R) = hAPP_f1566856189t_real(P,hAPP_r474017924t_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun,axiom,
    ! [P: fun_fu1716408315t_bool,Q: fun_re413263731l_bool,R: real] : hAPP_r847664445t_bool(hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,P),Q),R) = hAPP_f1173902867t_bool(P,hAPP_r1134773055l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_487,axiom,
    ! [P: fun_fu1635948867t_bool,Q: fun_re413263731l_bool,R: real] : hAPP_r279757445t_bool(hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,P),Q),R) = hAPP_f605995867t_bool(P,hAPP_r1134773055l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__488,axiom,
    ! [P: fun_fu2064567419t_real,Q: fun_in7315610l_real,R: int] : hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R) = hAPP_f766218899t_real(P,hAPP_i371647666l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__489,axiom,
    ! [P: fun_fu1138019779t_real,Q: fun_in7315610l_real,R: int] : hAPP_i447943416t_real(hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,P),Q),R) = hAPP_f1690286043t_real(P,hAPP_i371647666l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
    ! [P: fun_Pr1238001172nt_int,Q: fun_in2003698982nt_int,R: int] : hAPP_i1153444985nt_int(hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,P),Q),R) = hAPP_P621635040nt_int(P,hAPP_i1524277240nt_int(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
    ! [P: fun_Pr642042703nt_int,Q: fun_Pr480351984nt_int,R: product_prod_int_int] : hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,P),Q),R) = hAPP_f521271395nt_int(hAPP_P752050971nt_int(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__,axiom,
    ! [P: fun_bo1552468991nt_int,Q: fun_int_bool,R: int] : hAPP_i2104874254nt_int(hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,P),Q),R) = hAPP_b40753821nt_int(P,hAPP_int_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,axiom,
    ! [P: fun_fu587508068t_bool,Q: fun_in397122125l_bool,R: int] : hAPP_i1529485324t_bool(hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,P),Q),R) = hAPP_f727283836t_bool(P,hAPP_i1823918693l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL,axiom,
    ! [P: fun_fu1901564496t_bool,Q: fun_in1678813097l_bool,R: int] : hAPP_i1796122964t_bool(hAPP_f1902329768t_bool(hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,P),Q),R) = hAPP_f561022312t_bool(P,hAPP_i614188481l_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_,axiom,
    ! [P: fun_Pr755126262t_bool,Q: fun_Pr2144253564nt_int,R: product_prod_int_int] : hAPP_P989584703t_bool(hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,P),Q),R) = hAPP_f892584630t_bool(hAPP_P1400155668t_bool(P,R),hAPP_P1668131738nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__H,axiom,
    ! [P: fun_fu2112443261t_bool,Q: fun_Pr313471731t_bool,R: product_prod_int_int] : hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,P),Q),R) = hAPP_f237669397t_bool(P,hAPP_P989584703t_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__490,axiom,
    ! [P: fun_Pr1254492423t_bool,Q: fun_Pr313471731t_bool,R: product_prod_int_int] : hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,P),Q),R) = hAPP_f237669397t_bool(hAPP_P877703507t_bool(P,R),hAPP_P989584703t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__491,axiom,
    ! [P: fun_fu1230750243l_bool,Q: fun_re386138607t_bool,R: real] : hAPP_r1150855451l_bool(hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,P),Q),R) = hAPP_f1344929775l_bool(P,hAPP_r632163725t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__492,axiom,
    ! [P: fun_fu970548443l_bool,Q: fun_re2063775123t_bool,R: real] : hAPP_r285062775l_bool(hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,P),Q),R) = hAPP_f252743847l_bool(P,hAPP_r373117745t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal,axiom,
    ! [P: fun_fu888010351l_real,Q: fun_re1227808089l_real,R: real] : hAPP_r337325687l_real(hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,P),Q),R) = hAPP_f203520653l_real(P,hAPP_r337325687l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__493,axiom,
    ! [P: fun_fu1950887583l_real,Q: fun_re229243589l_real,R: real] : hAPP_r579619957l_real(hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,P),Q),R) = hAPP_f1591894897l_real(P,hAPP_r1250527377l_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p,axiom,
    ! [P: fun_re1233422975l_real,Q: fun_re1550493891l_real,R: real] : hAPP_r2019696015l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,P),Q),R) = hAPP_f1736103445l_real(hAPP_r224952523l_real(P,R),hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc___494,axiom,
    ! [P: fun_re1761459671l_real,Q: fun_re785671361l_real,R: real] : hAPP_r938550413l_real(hAPP_f1293550189l_real(hAPP_f183342685l_real(cOMBS_794971624l_real,P),Q),R) = hAPP_P1982799375l_real(hAPP_r931665397l_real(P,R),hAPP_r1195171167l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod,axiom,
    ! [P: fun_fu2104383023nt_int,Q: fun_in865885604nt_int,R: int] : hAPP_i1778081398nt_int(hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,P),Q),R) = hAPP_f465821005nt_int(P,hAPP_i1778081398nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int,axiom,
    ! [P: fun_fu156492935nt_int,Q: fun_in1111845413nt_int,R: int] : hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),Q),R) = hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int_495,axiom,
    ! [P: fun_fu1379407829nt_int,Q: fun_in1111845413nt_int,R: int] : hAPP_i1153444985nt_int(hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,P),Q),R) = hAPP_f1375227553nt_int(P,hAPP_i1584592887nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__I,axiom,
    ! [P: fun_fu686082665t_bool,Q: fun_Pr2144253564nt_int,R: product_prod_int_int] : hAPP_P2053772734t_bool(hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,P),Q),R) = hAPP_f291087797t_bool(P,hAPP_P1668131738nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_,axiom,
    ! [P: fun_bo1189210245l_real,Q: fun_real_bool,R: real] : hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R) = hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___496,axiom,
    ! [P: fun_fu2013528197nt_int,Q: fun_Pr2134412615nt_int,R: product_prod_int_int] : hAPP_P752050971nt_int(hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,P),Q),R) = hAPP_f11151005nt_int(P,hAPP_P2110489235nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc_497,axiom,
    ! [P: fun_fu201758811t_bool,Q: fun_Pr189909033l_bool,R: product_prod_int_int] : hAPP_P877703507t_bool(hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,P),Q),R) = hAPP_f513299955t_bool(P,hAPP_P921862901l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_,axiom,
    ! [P: fun_fu1269744292t_bool,Q: fun_re1598305103l_bool,R: real] : hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),Q),R) = hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__fun_,axiom,
    ! [P: fun_fu553683856t_bool,Q: fun_re1564496555l_bool,R: real] : hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R) = hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea_498,axiom,
    ! [P: fun_fu383125559l_real,Q: fun_re1550493891l_real,R: real] : hAPP_r712953929l_real(hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,P),Q),R) = hAPP_f800361423l_real(P,hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p_499,axiom,
    ! [P: fun_re1414183599l_real,Q: fun_re1550493891l_real,R: real] : hAPP_r1230691443l_real(hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,P),Q),R) = hAPP_f1293550189l_real(hAPP_r1043989325l_real(P,R),hAPP_r2019696015l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p,axiom,
    ! [P: fun_fu2102390523nt_int,Q: fun_in989261322nt_int,R: int] : hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),Q),R) = hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_,axiom,
    ! [P: fun_fu407119589t_bool,Q: fun_re1979725814t_bool,R: real] : hAPP_r1623423338t_bool(hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,P),Q),R) = hAPP_f472159229t_bool(P,hAPP_r1962736578t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_,axiom,
    ! [P: fun_fu1083576549t_bool,Q: fun_re506364222t_bool,R: real] : hAPP_r653725362t_bool(hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,P),Q),R) = hAPP_f729588221t_bool(P,hAPP_r1562208522t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mt,axiom,
    ! [P: fun_fu916648827t_bool,Q: fun_re1171866993t_bool,R: real] : hAPP_r1392521531t_bool(hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,P),Q),R) = hAPP_f1576199059t_bool(P,hAPP_r847664445t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mt,axiom,
    ! [P: fun_fu1861895035t_bool,Q: fun_re1091407545t_bool,R: real] : hAPP_r2052264067t_bool(hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,P),Q),R) = hAPP_f743790483t_bool(P,hAPP_r279757445t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_500,axiom,
    ! [P: fun_fu380422627l_real,Q: fun_re413263731l_bool,R: real] : hAPP_r794344869l_real(hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,P),Q),R) = hAPP_f1607917947l_real(P,hAPP_r1134773055l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_501,axiom,
    ! [P: fun_fu1698495721l_real,Q: fun_re665330775l_real,R: real] : hAPP_r712953929l_real(hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,P),Q),R) = hAPP_f1730463541l_real(P,hAPP_r579619957l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int__Oint,axiom,
    ! [P: fun_fu1280007140nt_int,Q: fun_in2124216673nt_int,R: int] : hAPP_i1836591008nt_int(hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,P),Q),R) = hAPP_f713949180nt_int(P,hAPP_i1153444985nt_int(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__f,axiom,
    ! [P: fun_fu581830891t_bool,Q: fun_Pr820506994t_bool,R: product_prod_int_int] : hAPP_P1400155668t_bool(hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,P),Q),R) = hAPP_f1325804733t_bool(P,hAPP_P2053772734t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real,axiom,
    ! [P: fun_fu2079587053l_real,Q: fun_re10870101l_real,R: real] : hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),Q),R) = hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real_502,axiom,
    ! [P: fun_fu209750539l_real,Q: fun_re69907033l_real,R: real] : hAPP_r1043989325l_real(hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,P),Q),R) = hAPP_f183342685l_real(P,hAPP_r794344869l_real(Q,R)) ).

%----Conjectures (1)
tff(conj_0,conjecture,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int)),zero_zero_int)) ).

%------------------------------------------------------------------------------